Theorem cbvcsbw 3053

Description: Version of cbvcsb 3054 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 2306	. . . 4 |- F/ y z e. C
3		cbvcsbw.2	. . . . 5 |- F/_ x D
4	3	nfcri 2306	. . . 4 |- F/ x z e. D
5		cbvcsbw.3	. . . . 5 |- ( x = y -> C = D )
6	5	eleq2d 2240	. . . 4 |- ( x = y -> ( z e. C <-> z e. D ) )
7	2, 4, 6	cbvsbcw 2982	. . 3 |- ( [. A / x ]. z e. C <-> [. A / y ]. z e. D )
8	7	abbii 2286	. 2 |- { z | [. A / x ]. z e. C } = { z | [. A / y ]. z e. D }
9		df-csb 3050	. 2 |- [_ A / x ]_ C = { z | [. A / x ]. z e. C }
10		df-csb 3050	. 2 |- [_ A / y ]_ D = { z | [. A / y ]. z e. D }
11	8, 9, 10	3eqtr4i 2201	1 |- [_ A / x ]_ C = [_ A / y ]_ D

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   {cab 2156   F/_wnfc 2299   [.wsbc 2955   [_csb 3049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-sbc 2956  df-csb 3050

This theorem is referenced by:  cbvprod  11521