Theorem bj-ssbcom3lem 32650

Description: Lemma for bj-ssbcom3 when setvar variables are disjoint. Remark: does not seem useful. (Contributed by BJ, 30-Dec-2020.)

Assertion

Ref	Expression
bj-ssbcom3lem	|- ([ t/ y]b[ y/ x]b ph <-> [ t/ y]b[ t/ x]b ph )

Distinct variable group:   x, y, t

Allowed substitution hints:   ph( x, y, t)

Proof of Theorem bj-ssbcom3lem

Step	Hyp	Ref	Expression
1		equequ2 1953	. . . . . . 7 |- ( y = t -> ( x = y <-> x = t ) )
2	1	imbi1d 331	. . . . . 6 |- ( y = t -> ( ( x = y -> ph ) <-> ( x = t -> ph ) ) )
3	2	pm5.74i 260	. . . . 5 |- ( ( y = t -> ( x = y -> ph ) ) <-> ( y = t -> ( x = t -> ph ) ) )
4	3	albii 1747	. . . 4 |- ( A. x ( y = t -> ( x = y -> ph ) ) <-> A. x ( y = t -> ( x = t -> ph ) ) )
5		19.21v 1868	. . . 4 |- ( A. x ( y = t -> ( x = y -> ph ) ) <-> ( y = t -> A. x ( x = y -> ph ) ) )
6		19.21v 1868	. . . 4 |- ( A. x ( y = t -> ( x = t -> ph ) ) <-> ( y = t -> A. x ( x = t -> ph ) ) )
7	4, 5, 6	3bitr3i 290	. . 3 |- ( ( y = t -> A. x ( x = y -> ph ) ) <-> ( y = t -> A. x ( x = t -> ph ) ) )
8	7	albii 1747	. 2 |- ( A. y ( y = t -> A. x ( x = y -> ph ) ) <-> A. y ( y = t -> A. x ( x = t -> ph ) ) )
9		bj-ssb1 32633	. . . 4 |- ([ y/ x]b ph <-> A. x ( x = y -> ph ) )
10	9	bj-ssbbii 32624	. . 3 |- ([ t/ y]b[ y/ x]b ph <-> [ t/ y]b A. x ( x = y -> ph ) )
11		bj-ssb1 32633	. . 3 |- ([ t/ y]b A. x ( x = y -> ph ) <-> A. y ( y = t -> A. x ( x = y -> ph ) ) )
12	10, 11	bitri 264	. 2 |- ([ t/ y]b[ y/ x]b ph <-> A. y ( y = t -> A. x ( x = y -> ph ) ) )
13		bj-ssb1 32633	. . . 4 |- ([ t/ x]b ph <-> A. x ( x = t -> ph ) )
14	13	bj-ssbbii 32624	. . 3 |- ([ t/ y]b[ t/ x]b ph <-> [ t/ y]b A. x ( x = t -> ph ) )
15		bj-ssb1 32633	. . 3 |- ([ t/ y]b A. x ( x = t -> ph ) <-> A. y ( y = t -> A. x ( x = t -> ph ) ) )
16	14, 15	bitri 264	. 2 |- ([ t/ y]b[ t/ x]b ph <-> A. y ( y = t -> A. x ( x = t -> ph ) ) )
17	8, 12, 16	3bitr4i 292	1 |- ([ t/ y]b[ y/ x]b ph <-> [ t/ y]b[ 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  ax-11 2034

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

This theorem is referenced by: (None)