Theorem sbceqg 3117

Description: Distribute proper substitution through an equality relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

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

Proof of Theorem sbceqg

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 3008	. . 3 |- ( z = A -> ( [ z / x ] B = C <-> [. A / x ]. B = C ) )
2		dfsbcq2 3008	. . . . 5 |- ( z = A -> ( [ z / x ] y e. B <-> [. A / x ]. y e. B ) )
3	2	abbidv 2325	. . . 4 |- ( z = A -> { y | [ z / x ] y e. B } = { y | [. A / x ]. y e. B } )
4		dfsbcq2 3008	. . . . 5 |- ( z = A -> ( [ z / x ] y e. C <-> [. A / x ]. y e. C ) )
5	4	abbidv 2325	. . . 4 |- ( z = A -> { y | [ z / x ] y e. C } = { y | [. A / x ]. y e. C } )
6	3, 5	eqeq12d 2222	. . 3 |- ( z = A -> ( { y | [ z / x ] y e. B } = { y | [ z / x ] y e. C } <-> { y | [. A / x ]. y e. B } = { y | [. A / x ]. y e. C } ) )
7		nfs1v 1968	. . . . . 6 |- F/ x [ z / x ] y e. B
8	7	nfab 2355	. . . . 5 |- F/_ x { y | [ z / x ] y e. B }
9		nfs1v 1968	. . . . . 6 |- F/ x [ z / x ] y e. C
10	9	nfab 2355	. . . . 5 |- F/_ x { y | [ z / x ] y e. C }
11	8, 10	nfeq 2358	. . . 4 |- F/ x { y | [ z / x ] y e. B } = { y | [ z / x ] y e. C }
12		sbab 2335	. . . . 5 |- ( x = z -> B = { y | [ z / x ] y e. B } )
13		sbab 2335	. . . . 5 |- ( x = z -> C = { y | [ z / x ] y e. C } )
14	12, 13	eqeq12d 2222	. . . 4 |- ( x = z -> ( B = C <-> { y | [ z / x ] y e. B } = { y | [ z / x ] y e. C } ) )
15	11, 14	sbie 1815	. . 3 |- ( [ z / x ] B = C <-> { y | [ z / x ] y e. B } = { y | [ z / x ] y e. C } )
16	1, 6, 15	vtoclbg 2839	. 2 |- ( A e. V -> ( [. A / x ]. B = C <-> { y | [. A / x ]. y e. B } = { y | [. A / x ]. y e. C } ) )
17		df-csb 3102	. . 3 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
18		df-csb 3102	. . 3 |- [_ A / x ]_ C = { y | [. A / x ]. y e. C }
19	17, 18	eqeq12i 2221	. 2 |- ( [_ A / x ]_ B = [_ A / x ]_ C <-> { y | [. A / x ]. y e. B } = { y | [. A / x ]. y e. C } )
20	16, 19	bitr4di 198	1 |- ( A e. V -> ( [. A / x ]. B = C <-> [_ A / x ]_ B = [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   [wsb 1786   e. wcel 2178   {cab 2193   [.wsbc 3005   [_csb 3101

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-sbc 3006  df-csb 3102

This theorem is referenced by:  sbcne12g  3119  sbceq1g  3121  sbceq2g  3123  sbcfng  5443  swrdspsleq  11158  fprodmodd  12067