Theorem ax5el 34222

Description: Theorem to add distinct quantifier to atomic formula. This theorem demonstrates the induction basis for ax-5 1839 considered as a metatheorem.) (Contributed by NM, 22-Jun-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax5el	|- ( x e. y -> A. z x e. y )

Distinct variable groups:   x, z   y, z

Proof of Theorem ax5el

Step	Hyp	Ref	Expression
1		ax-c14 34176	. 2 |- ( -. A. z z = x -> ( -. A. z z = y -> ( x e. y -> A. z x e. y ) ) )
2		ax-c16 34177	. 2 |- ( A. z z = x -> ( x e. y -> A. z x e. y ) )
3		ax-c16 34177	. 2 |- ( A. z z = y -> ( x e. y -> A. z x e. y ) )
4	1, 2, 3	pm2.61ii 177	1 |- ( x e. y -> A. z x e. y )

Syntax hints:   -> wi 4   A.wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-c14 34176  ax-c16 34177

This theorem is referenced by:  dveel2ALT  34224