Theorem ax5eq

Description: Theorem to add distinct quantifier to atomic formula. (This theorem demonstrates the induction basis for ax-5 considered as a metatheorem. Do not use it for later proofs - use ax-5 instead, to avoid reference to the redundant axiom ax-c16 .) (Contributed by NM, 10-Jan-1993) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	ax5eq	|- ( x = y -> A. z x = y )

Proof

Step	Hyp	Ref	Expression
1		ax-c9	|- ( -. A. z z = x -> ( -. A. z z = y -> ( x = y -> A. z x = y ) ) )
2		ax-c16	|- ( A. z z = x -> ( x = y -> A. z x = y ) )
3		ax-c16	|- ( A. z z = y -> ( x = y -> A. z x = y ) )
4	1 2 3	pm2.61ii	|- ( x = y -> A. z x = y )