Theorem bj-ssblem2

Description: An instance of ax-11 proved without it. The converse may not be provable without ax-11 (since using alcomiw would require a DV on ph , x , which defeats the purpose). (Contributed by BJ, 22-Dec-2020) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-ssblem2	|- ( A. x A. y ( y = t -> ( x = y -> ph ) ) -> A. y A. x ( y = t -> ( x = y -> ph ) ) )

Proof

Step	Hyp	Ref	Expression
1		equequ1	|- ( y = z -> ( y = t <-> z = t ) )
2		equequ2	|- ( y = z -> ( x = y <-> x = z ) )
3	2	imbi1d	|- ( y = z -> ( ( x = y -> ph ) <-> ( x = z -> ph ) ) )
4	1 3	imbi12d	|- ( y = z -> ( ( y = t -> ( x = y -> ph ) ) <-> ( z = t -> ( x = z -> ph ) ) ) )
5	4	alcomiw	|- ( A. x A. y ( y = t -> ( x = y -> ph ) ) -> A. y A. x ( y = t -> ( x = y -> ph ) ) )