Theorem ax17el 1354

Description: Theorem to add distinct quantifier to atomic formula. This theorem demonstrates the induction basis for ax-17 968 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 1353	. 2 |- (-. A.z z = x -> (-. A.z z = y -> (x e. y -> A.z x e. y)))
2		ax-16 1206	. 2 |- (A.z z = x -> (x e. y -> A.z x e. y))
3		ax-16 1206	. 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 951   e. wcel 955

This theorem is referenced by:  dveel2ALT 1355

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-16 1206  ax-15 1353

Copyright terms: Public domain