Theorem bj-sbceqgALT 32897

Description: Distribute proper substitution through an equality relation. Alternate proof of sbceqg 3984. (Contributed by BJ, 6-Oct-2018.) Proof modification is discouraged to avoid using sbceqg 3984, but "minimize */except sbceqg" is ok. (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-sbceqgALT	|- ( A e. V -> ( [. A / x ]. B = C <-> [_ A / x ]_ B = [_ A / x ]_ C ) )

Proof of Theorem bj-sbceqgALT

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfcleq 2616	. . . . . 6 |- ( B = C <-> A. y ( y e. B <-> y e. C ) )
2	1	sbcth 3450	. . . . 5 |- ( A e. V -> [. A / x ]. ( B = C <-> A. y ( y e. B <-> y e. C ) ) )
3		sbcbig 3480	. . . . 5 |- ( A e. V -> ( [. A / x ]. ( B = C <-> A. y ( y e. B <-> y e. C ) ) <-> ( [. A / x ]. B = C <-> [. A / x ]. A. y ( y e. B <-> y e. C ) ) ) )
4	2, 3	mpbid 222	. . . 4 |- ( A e. V -> ( [. A / x ]. B = C <-> [. A / x ]. A. y ( y e. B <-> y e. C ) ) )
5		sbcal 3485	. . . 4 |- ( [. A / x ]. A. y ( y e. B <-> y e. C ) <-> A. y [. A / x ]. ( y e. B <-> y e. C ) )
6	4, 5	syl6bb 276	. . 3 |- ( A e. V -> ( [. A / x ]. B = C <-> A. y [. A / x ]. ( y e. B <-> y e. C ) ) )
7		sbcbig 3480	. . . 4 |- ( A e. V -> ( [. A / x ]. ( y e. B <-> y e. C ) <-> ( [. A / x ]. y e. B <-> [. A / x ]. y e. C ) ) )
8	7	albidv 1849	. . 3 |- ( A e. V -> ( A. y [. A / x ]. ( y e. B <-> y e. C ) <-> A. y ( [. A / x ]. y e. B <-> [. A / x ]. y e. C ) ) )
9		sbcel2 3989	. . . . . 6 |- ( [. A / x ]. y e. B <-> y e. [_ A / x ]_ B )
10	9	a1i 11	. . . . 5 |- ( A e. V -> ( [. A / x ]. y e. B <-> y e. [_ A / x ]_ B ) )
11		sbcel2 3989	. . . . . 6 |- ( [. A / x ]. y e. C <-> y e. [_ A / x ]_ C )
12	11	a1i 11	. . . . 5 |- ( A e. V -> ( [. A / x ]. y e. C <-> y e. [_ A / x ]_ C ) )
13	10, 12	bibi12d 335	. . . 4 |- ( A e. V -> ( ( [. A / x ]. y e. B <-> [. A / x ]. y e. C ) <-> ( y e. [_ A / x ]_ B <-> y e. [_ A / x ]_ C ) ) )
14	13	albidv 1849	. . 3 |- ( A e. V -> ( A. y ( [. A / x ]. y e. B <-> [. A / x ]. y e. C ) <-> A. y ( y e. [_ A / x ]_ B <-> y e. [_ A / x ]_ C ) ) )
15	6, 8, 14	3bitrd 294	. 2 |- ( A e. V -> ( [. A / x ]. B = C <-> A. y ( y e. [_ A / x ]_ B <-> y e. [_ A / x ]_ C ) ) )
16		dfcleq 2616	. 2 |- ( [_ A / x ]_ B = [_ A / x ]_ C <-> A. y ( y e. [_ A / x ]_ B <-> y e. [_ A / x ]_ C ) )
17	15, 16	syl6bbr 278	1 |- ( A e. V -> ( [. A / x ]. B = C <-> [_ A / x ]_ B = [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   = wceq 1483   e. wcel 1990   [.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)