Theorem cbvcsbw 3063

Description: Version of cbvcsb 3064 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.)

Hypotheses

Ref	Expression
cbvcsbw.1	|- F/_ y C
cbvcsbw.2	|- F/_ x D
cbvcsbw.3	|- ( x = y -> C = D )

Assertion

Ref	Expression
cbvcsbw	|- [_ A / x ]_ C = [_ A / y ]_ D

Distinct variable group:   x, y

Allowed substitution hints:   A( x, y)   C( x, y)   D( x, y)

Proof of Theorem cbvcsbw

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvcsbw.1	. . . . 5 |- F/_ y C
2	1	nfcri 2313	. . . 4 |- F/ y z e. C
3		cbvcsbw.2	. . . . 5 |- F/_ x D
4	3	nfcri 2313	. . . 4 |- F/ x z e. D
5		cbvcsbw.3	. . . . 5 |- ( x = y -> C = D )
6	5	eleq2d 2247	. . . 4 |- ( x = y -> ( z e. C <-> z e. D ) )
7	2, 4, 6	cbvsbcw 2992	. . 3 |- ( [. A / x ]. z e. C <-> [. A / y ]. z e. D )
8	7	abbii 2293	. 2 |- { z | [. A / x ]. z e. C } = { z | [. A / y ]. z e. D }
9		df-csb 3060	. 2 |- [_ A / x ]_ C = { z | [. A / x ]. z e. C }
10		df-csb 3060	. 2 |- [_ A / y ]_ D = { z | [. A / y ]. z e. D }
11	8, 9, 10	3eqtr4i 2208	1 |- [_ A / x ]_ C = [_ A / y ]_ D

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   {cab 2163   F/_wnfc 2306   [.wsbc 2964   [_csb 3059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-sbc 2965  df-csb 3060

This theorem is referenced by:  cbvprod  11568