Theorem cbvcsbw 3131

Description: Version of cbvcsb 3132 with a disjoint variable condition. (Contributed by GG, 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 2368	. . . 4 |- F/ y z e. C
3		cbvcsbw.2	. . . . 5 |- F/_ x D
4	3	nfcri 2368	. . . 4 |- F/ x z e. D
5		cbvcsbw.3	. . . . 5 |- ( x = y -> C = D )
6	5	eleq2d 2301	. . . 4 |- ( x = y -> ( z e. C <-> z e. D ) )
7	2, 4, 6	cbvsbcw 3059	. . 3 |- ( [. A / x ]. z e. C <-> [. A / y ]. z e. D )
8	7	abbii 2347	. 2 |- { z | [. A / x ]. z e. C } = { z | [. A / y ]. z e. D }
9		df-csb 3128	. 2 |- [_ A / x ]_ C = { z | [. A / x ]. z e. C }
10		df-csb 3128	. 2 |- [_ A / y ]_ D = { z | [. A / y ]. z e. D }
11	8, 9, 10	3eqtr4i 2262	1 |- [_ A / x ]_ C = [_ A / y ]_ D

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   {cab 2217   F/_wnfc 2361   [.wsbc 3031   [_csb 3127

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-sbc 3032  df-csb 3128

This theorem is referenced by:  cbvprod  12118