Theorem ax5el

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

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

Proof

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