Theorem equs5

Description: Lemma used in proofs of substitution properties. If there is a disjoint variable condition on x , y , then sb56 can be used instead; if y is not free in ph , then equs45f can be used. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 14-May-1993) (Revised by BJ, 1-Oct-2018) (New usage is discouraged.)

	Ref	Expression
Assertion	equs5	|- ( -. A. x x = y -> ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) ) )

Proof

Step	Hyp	Ref	Expression
1		nfna1	|- F/ x -. A. x x = y
2		nfa1	|- F/ x A. x ( x = y -> ph )
3		axc15	|- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
4	3	impd	|- ( -. A. x x = y -> ( ( x = y /\ ph ) -> A. x ( x = y -> ph ) ) )
5	1 2 4	exlimd	|- ( -. A. x x = y -> ( E. x ( x = y /\ ph ) -> A. x ( x = y -> ph ) ) )
6		equs4	|- ( A. x ( x = y -> ph ) -> E. x ( x = y /\ ph ) )
7	5 6	impbid1	|- ( -. A. x x = y -> ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) ) )