Theorem bj-ssblem2 32631

Description: An instance of ax-11 2034 proved without it. The converse may not be provable without ax-11 2034 (since using alcomiw 1971 would require a DV on ph , x, which defeats the purpose). (Contributed by BJ, 22-Dec-2020.)

Assertion

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

Distinct variable groups:   x, y, t   ph, y

Allowed substitution hints:   ph( x, t)

Proof of Theorem bj-ssblem2

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equequ1 1952	. . 3 |- ( y = z -> ( y = t <-> z = t ) )
2		equequ2 1953	. . . 4 |- ( y = z -> ( x = y <-> x = z ) )
3	2	imbi1d 331	. . 3 |- ( y = z -> ( ( x = y -> ph ) <-> ( x = z -> ph ) ) )
4	1, 3	imbi12d 334	. 2 |- ( y = z -> ( ( y = t -> ( x = y -> ph ) ) <-> ( z = t -> ( x = z -> ph ) ) ) )
5	4	alcomiw 1971	1 |- ( A. x A. y ( y = t -> ( x = y -> ph ) ) -> A. y A. x ( y = t -> ( x = y -> ph ) ) )

Syntax hints:   -> wi 4   A.wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by:  bj-ssb1a  32632