Theorem equs5a

Description: A property related to substitution that unlike equs5 does not require a distinctor antecedent. Usage of this theorem is discouraged because it depends on ax-13 . This proof uses ax12 , see equs5aALT for an alternative one using ax-12 but not ax13 . Usage of the weaker equs5av is preferred, which uses ax12v2 , but not ax-13 . (Contributed by NM, 2-Feb-2007) (New usage is discouraged.)

	Ref	Expression
Assertion	equs5a	|- ( E. x ( x = y /\ A. y ph ) -> A. x ( x = y -> ph ) )

Proof

Step	Hyp	Ref	Expression
1		nfa1	|- F/ x A. x ( x = y -> ph )
2		ax12	|- ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) )
3	2	imp	|- ( ( x = y /\ A. y ph ) -> A. x ( x = y -> ph ) )
4	1 3	exlimi	|- ( E. x ( x = y /\ A. y ph ) -> A. x ( x = y -> ph ) )