Theorem bj-ssbequ2 32643

Description: Note that ax-12 2047 is used only via sp 2053. See sbequ2 1882 and stdpc7 1958. (Contributed by BJ, 22-Dec-2020.)

Assertion

Ref	Expression
bj-ssbequ2	|- ( x = t -> ([ t/ x]b ph -> ph ) )

Proof of Theorem bj-ssbequ2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ssb 32620	. . 3 |- ([ t/ x]b ph <-> A. y ( y = t -> A. x ( x = y -> ph ) ) )
2		sp 2053	. . . . . 6 |- ( A. x ( x = y -> ph ) -> ( x = y -> ph ) )
3	2	imim2i 16	. . . . 5 |- ( ( y = t -> A. x ( x = y -> ph ) ) -> ( y = t -> ( x = y -> ph ) ) )
4	3	alimi 1739	. . . 4 |- ( A. y ( y = t -> A. x ( x = y -> ph ) ) -> A. y ( y = t -> ( x = y -> ph ) ) )
5		pm3.31 461	. . . . 5 |- ( ( y = t -> ( x = y -> ph ) ) -> ( ( y = t /\ x = y ) -> ph ) )
6	5	alimi 1739	. . . 4 |- ( A. y ( y = t -> ( x = y -> ph ) ) -> A. y ( ( y = t /\ x = y ) -> ph ) )
7		19.23v 1902	. . . . 5 |- ( A. y ( ( y = t /\ x = y ) -> ph ) <-> ( E. y ( y = t /\ x = y ) -> ph ) )
8		equviniva 1960	. . . . . . 7 |- ( x = t -> E. y ( x = y /\ t = y ) )
9		biid 251	. . . . . . . . . . . 12 |- ( x = y <-> x = y )
10		equcom 1945	. . . . . . . . . . . 12 |- ( t = y <-> y = t )
11	9, 10	anbi12ci 734	. . . . . . . . . . 11 |- ( ( x = y /\ t = y ) <-> ( y = t /\ x = y ) )
12	11	biimpi 206	. . . . . . . . . 10 |- ( ( x = y /\ t = y ) -> ( y = t /\ x = y ) )
13	12	eximi 1762	. . . . . . . . 9 |- ( E. y ( x = y /\ t = y ) -> E. y ( y = t /\ x = y ) )
14		pm3.35 611	. . . . . . . . 9 |- ( ( E. y ( y = t /\ x = y ) /\ ( E. y ( y = t /\ x = y ) -> ph ) ) -> ph )
15	13, 14	sylan 488	. . . . . . . 8 |- ( ( E. y ( x = y /\ t = y ) /\ ( E. y ( y = t /\ x = y ) -> ph ) ) -> ph )
16	15	ancoms 469	. . . . . . 7 |- ( ( ( E. y ( y = t /\ x = y ) -> ph ) /\ E. y ( x = y /\ t = y ) ) -> ph )
17	8, 16	sylan2 491	. . . . . 6 |- ( ( ( E. y ( y = t /\ x = y ) -> ph ) /\ x = t ) -> ph )
18	17	ex 450	. . . . 5 |- ( ( E. y ( y = t /\ x = y ) -> ph ) -> ( x = t -> ph ) )
19	7, 18	sylbi 207	. . . 4 |- ( A. y ( ( y = t /\ x = y ) -> ph ) -> ( x = t -> ph ) )
20	4, 6, 19	3syl 18	. . 3 |- ( A. y ( y = t -> A. x ( x = y -> ph ) ) -> ( x = t -> ph ) )
21	1, 20	sylbi 207	. 2 |- ([ t/ x]b ph -> ( x = t -> ph ) )
22	21	com12 32	1 |- ( x = t -> ([ t/ x]b ph -> ph ) )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   E.wex 1704  [wssb 32619

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  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-ssb 32620

This theorem is referenced by:  bj-ssbid2  32645