Theorem sbcel12g 3064

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

Assertion

Ref	Expression
sbcel12g	|- ( A e. V -> ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C ) )

Proof of Theorem sbcel12g

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

Step	Hyp	Ref	Expression
1		dfsbcq2 2958	. . 3 |- ( z = A -> ( [ z / x ] B e. C <-> [. A / x ]. B e. C ) )
2		dfsbcq2 2958	. . . . 5 |- ( z = A -> ( [ z / x ] y e. B <-> [. A / x ]. y e. B ) )
3	2	abbidv 2288	. . . 4 |- ( z = A -> { y | [ z / x ] y e. B } = { y | [. A / x ]. y e. B } )
4		dfsbcq2 2958	. . . . 5 |- ( z = A -> ( [ z / x ] y e. C <-> [. A / x ]. y e. C ) )
5	4	abbidv 2288	. . . 4 |- ( z = A -> { y | [ z / x ] y e. C } = { y | [. A / x ]. y e. C } )
6	3, 5	eleq12d 2241	. . 3 |- ( z = A -> ( { y | [ z / x ] y e. B } e. { y | [ z / x ] y e. C } <-> { y | [. A / x ]. y e. B } e. { y | [. A / x ]. y e. C } ) )
7		nfs1v 1932	. . . . . 6 |- F/ x [ z / x ] y e. B
8	7	nfab 2317	. . . . 5 |- F/_ x { y | [ z / x ] y e. B }
9		nfs1v 1932	. . . . . 6 |- F/ x [ z / x ] y e. C
10	9	nfab 2317	. . . . 5 |- F/_ x { y | [ z / x ] y e. C }
11	8, 10	nfel 2321	. . . 4 |- F/ x { y | [ z / x ] y e. B } e. { y | [ z / x ] y e. C }
12		sbab 2298	. . . . 5 |- ( x = z -> B = { y | [ z / x ] y e. B } )
13		sbab 2298	. . . . 5 |- ( x = z -> C = { y | [ z / x ] y e. C } )
14	12, 13	eleq12d 2241	. . . 4 |- ( x = z -> ( B e. C <-> { y | [ z / x ] y e. B } e. { y | [ z / x ] y e. C } ) )
15	11, 14	sbie 1784	. . 3 |- ( [ z / x ] B e. C <-> { y | [ z / x ] y e. B } e. { y | [ z / x ] y e. C } )
16	1, 6, 15	vtoclbg 2791	. 2 |- ( A e. V -> ( [. A / x ]. B e. C <-> { y | [. A / x ]. y e. B } e. { y | [. A / x ]. y e. C } ) )
17		df-csb 3050	. . 3 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
18		df-csb 3050	. . 3 |- [_ A / x ]_ C = { y | [. A / x ]. y e. C }
19	17, 18	eleq12i 2238	. 2 |- ( [_ A / x ]_ B e. [_ A / x ]_ C <-> { y | [. A / x ]. y e. B } e. { y | [. A / x ]. y e. C } )
20	16, 19	bitr4di 197	1 |- ( A e. V -> ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   [wsb 1755   e. wcel 2141   {cab 2156   [.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-v 2732  df-sbc 2956  df-csb 3050

This theorem is referenced by:  sbcnel12g  3066  sbcel1g  3068  sbcel2g  3070  sbccsb2g  3079  ixpsnval  6679