Theorem bj-ssbequ 32629

Description: Equality property for substitution, from Tarski's system. Compare sbequ 2376. (Contributed by BJ, 30-Dec-2020.)

Assertion

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

Proof of Theorem bj-ssbequ

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equequ2 1953	. . . 4 |- ( s = t -> ( y = s <-> y = t ) )
2	1	imbi1d 331	. . 3 |- ( s = t -> ( ( y = s -> A. x ( x = y -> ph ) ) <-> ( y = t -> A. x ( x = y -> ph ) ) ) )
3	2	albidv 1849	. 2 |- ( s = t -> ( A. y ( y = s -> A. x ( x = y -> ph ) ) <-> A. y ( y = t -> A. x ( x = y -> ph ) ) ) )
4		df-ssb 32620	. 2 |- ([ s/ x]b ph <-> A. y ( y = s -> A. x ( x = y -> ph ) ) )
5		df-ssb 32620	. 2 |- ([ t/ x]b ph <-> A. y ( y = t -> A. x ( x = y -> ph ) ) )
6	3, 4, 5	3bitr4g 303	1 |- ( s = t -> ([ s/ x]b ph <-> [ t/ x]b ph ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481  [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

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

This theorem is referenced by: (None)