Theorem ax17el 1403

Description: Theorem to add distinct quantifier to atomic formula. This theorem demonstrates the induction basis for ax-17 1012 considered as a metatheorem.)

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 1402	. 2 |- (-. A.z z = x -> (-. A.z z = y -> (x e. y -> A.z x e. y)))
2		ax-16 1252	. 2 |- (A.z z = x -> (x e. y -> A.z x e. y))
3		ax-16 1252	. 2 |- (A.z z = y -> (x e. y -> A.z x e. y))
4	1, 2, 3	pm2.61ii 136	1 |- (x e. y -> A.z x e. y)

Syntax hints:   -> wi 3  A.wal 995   e. wcel 999

This theorem is referenced by:  dveel2ALT 1404

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-16 1252  ax-15 1402

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