Theorem bj-sbeq 32896

Description: Distribute proper substitution through an equality relation. (See sbceqg 3984). (Contributed by BJ, 6-Oct-2018.)

Assertion

Ref	Expression
bj-sbeq	|- ( [ y / x ] A = B <-> [_ y / x ]_ A = [_ y / x ]_ B )

Proof of Theorem bj-sbeq

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfcleq 2616	. . . . 5 |- ( A = B <-> A. z ( z e. A <-> z e. B ) )
2	1	sbbii 1887	. . . 4 |- ( [ y / x ] A = B <-> [ y / x ] A. z ( z e. A <-> z e. B ) )
3		sbsbc 3439	. . . 4 |- ( [ y / x ] A. z ( z e. A <-> z e. B ) <-> [. y / x ]. A. z ( z e. A <-> z e. B ) )
4		sbcal 3485	. . . 4 |- ( [. y / x ]. A. z ( z e. A <-> z e. B ) <-> A. z [. y / x ]. ( z e. A <-> z e. B ) )
5	2, 3, 4	3bitri 286	. . 3 |- ( [ y / x ] A = B <-> A. z [. y / x ]. ( z e. A <-> z e. B ) )
6		vex 3203	. . . . 5 |- y e. _V
7		sbcbig 3480	. . . . 5 |- ( y e. _V -> ( [. y / x ]. ( z e. A <-> z e. B ) <-> ( [. y / x ]. z e. A <-> [. y / x ]. z e. B ) ) )
8	6, 7	ax-mp 5	. . . 4 |- ( [. y / x ]. ( z e. A <-> z e. B ) <-> ( [. y / x ]. z e. A <-> [. y / x ]. z e. B ) )
9	8	albii 1747	. . 3 |- ( A. z [. y / x ]. ( z e. A <-> z e. B ) <-> A. z ( [. y / x ]. z e. A <-> [. y / x ]. z e. B ) )
10		sbcel2 3989	. . . . 5 |- ( [. y / x ]. z e. A <-> z e. [_ y / x ]_ A )
11		sbcel2 3989	. . . . 5 |- ( [. y / x ]. z e. B <-> z e. [_ y / x ]_ B )
12	10, 11	bibi12i 329	. . . 4 |- ( ( [. y / x ]. z e. A <-> [. y / x ]. z e. B ) <-> ( z e. [_ y / x ]_ A <-> z e. [_ y / x ]_ B ) )
13	12	albii 1747	. . 3 |- ( A. z ( [. y / x ]. z e. A <-> [. y / x ]. z e. B ) <-> A. z ( z e. [_ y / x ]_ A <-> z e. [_ y / x ]_ B ) )
14	5, 9, 13	3bitri 286	. 2 |- ( [ y / x ] A = B <-> A. z ( z e. [_ y / x ]_ A <-> z e. [_ y / x ]_ B ) )
15		dfcleq 2616	. 2 |- ( [_ y / x ]_ A = [_ y / x ]_ B <-> A. z ( z e. [_ y / x ]_ A <-> z e. [_ y / x ]_ B ) )
16	14, 15	bitr4i 267	1 |- ( [ y / x ] A = B <-> [_ y / x ]_ A = [_ y / x ]_ B )

Syntax hints:   <-> wb 196   A.wal 1481   = wceq 1483   [wsb 1880   e. wcel 1990   _Vcvv 3200   [.wsbc 3435   [_csb 3533

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-nul 3916

This theorem is referenced by: (None)