Theorem ax12b

Description: A bidirectional version of axc15 . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 30-Jun-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	ax12b	|- ( ( -. A. x x = y /\ x = y ) -> ( ph <-> A. x ( x = y -> ph ) ) )

Proof

Step	Hyp	Ref	Expression
1		axc15	|- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
2	1	imp	|- ( ( -. A. x x = y /\ x = y ) -> ( ph -> A. x ( x = y -> ph ) ) )
3		sp	|- ( A. x ( x = y -> ph ) -> ( x = y -> ph ) )
4	3	com12	|- ( x = y -> ( A. x ( x = y -> ph ) -> ph ) )
5	4	adantl	|- ( ( -. A. x x = y /\ x = y ) -> ( A. x ( x = y -> ph ) -> ph ) )
6	2 5	impbid	|- ( ( -. A. x x = y /\ x = y ) -> ( ph <-> A. x ( x = y -> ph ) ) )