Theorem cbvcsbw 3097

Description: Version of cbvcsb 3098 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 2342	. . . 4 |- F/ y z e. C
3		cbvcsbw.2	. . . . 5 |- F/_ x D
4	3	nfcri 2342	. . . 4 |- F/ x z e. D
5		cbvcsbw.3	. . . . 5 |- ( x = y -> C = D )
6	5	eleq2d 2275	. . . 4 |- ( x = y -> ( z e. C <-> z e. D ) )
7	2, 4, 6	cbvsbcw 3026	. . 3 |- ( [. A / x ]. z e. C <-> [. A / y ]. z e. D )
8	7	abbii 2321	. 2 |- { z | [. A / x ]. z e. C } = { z | [. A / y ]. z e. D }
9		df-csb 3094	. 2 |- [_ A / x ]_ C = { z | [. A / x ]. z e. C }
10		df-csb 3094	. 2 |- [_ A / y ]_ D = { z | [. A / y ]. z e. D }
11	8, 9, 10	3eqtr4i 2236	1 |- [_ A / x ]_ C = [_ A / y ]_ D

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2176   {cab 2191   F/_wnfc 2335   [.wsbc 2998   [_csb 3093

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-sbc 2999  df-csb 3094

This theorem is referenced by:  cbvprod  11869