Theorem bj-ssbeq 32627

Description: Substitution in an equality, disjoint variables case. Uses only ax-1--6. It might be shorter to prove the result about composition of two substitutions and prove bj-ssbeq 32627 first with a DV on x,t, and then in the general case. (Contributed by BJ, 22-Dec-2020.)

Assertion

Ref	Expression
bj-ssbeq	|- ([ t/ x]b y = z <-> y = z )

Distinct variable groups:   x, y   x, z

Proof of Theorem bj-ssbeq

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ssb 32620	. 2 |- ([ t/ x]b y = z <-> A. u ( u = t -> A. x ( x = u -> y = z ) ) )
2		19.23v 1902	. . . . . 6 |- ( A. x ( x = u -> y = z ) <-> ( E. x x = u -> y = z ) )
3		ax6ev 1890	. . . . . . . 8 |- E. x x = u
4		pm2.27 42	. . . . . . . 8 |- ( E. x x = u -> ( ( E. x x = u -> y = z ) -> y = z ) )
5	3, 4	ax-mp 5	. . . . . . 7 |- ( ( E. x x = u -> y = z ) -> y = z )
6		ax-1 6	. . . . . . 7 |- ( y = z -> ( E. x x = u -> y = z ) )
7	5, 6	impbii 199	. . . . . 6 |- ( ( E. x x = u -> y = z ) <-> y = z )
8	2, 7	bitri 264	. . . . 5 |- ( A. x ( x = u -> y = z ) <-> y = z )
9	8	imbi2i 326	. . . 4 |- ( ( u = t -> A. x ( x = u -> y = z ) ) <-> ( u = t -> y = z ) )
10	9	albii 1747	. . 3 |- ( A. u ( u = t -> A. x ( x = u -> y = z ) ) <-> A. u ( u = t -> y = z ) )
11		19.23v 1902	. . . 4 |- ( A. u ( u = t -> y = z ) <-> ( E. u u = t -> y = z ) )
12		ax6ev 1890	. . . . . 6 |- E. u u = t
13		pm2.27 42	. . . . . 6 |- ( E. u u = t -> ( ( E. u u = t -> y = z ) -> y = z ) )
14	12, 13	ax-mp 5	. . . . 5 |- ( ( E. u u = t -> y = z ) -> y = z )
15		ax-1 6	. . . . 5 |- ( y = z -> ( E. u u = t -> y = z ) )
16	14, 15	impbii 199	. . . 4 |- ( ( E. u u = t -> y = z ) <-> y = z )
17	11, 16	bitri 264	. . 3 |- ( A. u ( u = t -> y = z ) <-> y = z )
18	10, 17	bitri 264	. 2 |- ( A. u ( u = t -> A. x ( x = u -> y = z ) ) <-> y = z )
19	1, 18	bitri 264	1 |- ([ t/ x]b y = z <-> y = z )

Syntax hints:   -> wi 4   <-> wb 196   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

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

This theorem is referenced by: (None)