Snapcaster Mage and Phyrexian Mana Costs

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I know that Snapcaster Mage allows one to replay an instant or sorcery for its converted mana cost, and because of this one cannot cast Archive Trap for its trap cost. Does this mean that one could not play Mental Misstep with Snapcaster by being able to pay life for its mana cost?

EDIT: also, can I only cast a sorcery with the mage during my turn?
Snapcaster mage has nothing to do with converted mana cost. It looks at the mana cost, which includes all the symbols. Converted mana cost is just a number. Archive trap has an alternative cost, and flashback is already an alternative cost, so you can't use both. You can only use one alternative cost when casting a spell.

You can in fact pay life for phyrexian symbols in spells you're flashing back with snapcaster mage.

Snapcaster mage doesn't give the spell flash, just flashback, so you can only cast a sorcery whenever you could normally cast a sorcery. It's no different than any other sorcery with flashback.
All Generalizations are Bad
phyrexian mana is not strictly speaking an alternative cost like the from the trap or something like Force of Will
you can pay phyrexian mana in the flashback cost with mana or 2 life as usual

the snapcaster mage doesn't change timing restrictions, if you give a sorcery flashback you can only cast it any time you normally can cast a sorcery.
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I figured as much about the sorcery but I was not sure about the phyrexian mana symbols. Thanks!

Mana cost of Archive Trap: . Converted mana cost of Archive Trap: 5.

A mana cost is a group of mana symbols displayed in the top right corner of a card (except for those silly Future Sight frames, but disregard those, where it's on the left). A card's converted mana cost is an integer value. Even cards without mana costs (like lands for example) have converted mana costs. You'll note that Snapcaster Mage mentions the targeted spell's mana cost (not converted mana cost). Thus if any of the symbols in the cost give you a choice of how they can be paid, like phyrexian or hybrid mana, you get to choose how to pay them. Thus you can opt to pay the phyrexian bits of mana costs with life. This is an intrinsic property of phyrexian mana and a completely different thing than alternate casting costs, which are always spelled out in the text box.

For an example of things that care about a card's converted mana cost instead, take a look at spells with the Cascade mechanic.

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