Theorem ax17el 1196

Description: Theorem to add distinct quantifier to atomic formula. (This theorem demonstrates the induction basis for ax-17 1190 considered as a metatheorem. Do not use it for later proofs - use ax-17 1190 instead, to avoid reference to the redundant axioms ax-15 1109 and ax-16 1194.)

Assertion

Ref	Expression
ax17el	|- (x e. y -> A.z x e. y)

Distinct variable groups:   x,z   y,z

Proof of Theorem ax17el

Step	Hyp	Ref	Expression
1		ax-15 1109	. 2 |- (-. A.z z = x -> (-. A.z z = y -> (x e. y -> A.z x e. y)))
2		ax-16 1194	. 2 |- (A.z z = x -> (x e. y -> A.z x e. y))
3		ax-16 1194	. 2 |- (A.z z = y -> (x e. y -> A.z x e. y))
4	1, 2, 3	pm2.61ii 130	1 |- (x e. y -> A.z x e. y)

Syntax hints:   -> wi 3  A.wal 950   e. wcel 1105

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-15 1109  ax-16 1194

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