Theorem sbcel12g 3152

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 3050	. . 3 |- (z = A -> ([z / x]B e. C <-> [.A / x ].B e. C))
2		dfsbcq2 3050	. . . . 5 |- (z = A -> ([z / x]y e. B <-> [.A / x ].y e. B))
3	2	abbidv 2468	. . . 4 |- (z = A -> {y | [z / x]y e. B} = {y | [.A / x ].y e. B})
4		dfsbcq2 3050	. . . . 5 |- (z = A -> ([z / x]y e. C <-> [.A / x ].y e. C))
5	4	abbidv 2468	. . . 4 |- (z = A -> {y | [z / x]y e. C} = {y | [.A / x ].y e. C})
6	3, 5	eleq12d 2421	. . 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 2106	. . . . . 6 |- F/x[z / x]y e. B
8	7	nfab 2494	. . . . 5 |- F/_x{y | [z / x]y e. B}
9		nfs1v 2106	. . . . . 6 |- F/x[z / x]y e. C
10	9	nfab 2494	. . . . 5 |- F/_x{y | [z / x]y e. C}
11	8, 10	nfel 2498	. . . 4 |- F/x{y | [z / x]y e. B} e. {y | [z / x]y e. C}
12		sbab 2476	. . . . 5 |- (x = z -> B = {y | [z / x]y e. B})
13		sbab 2476	. . . . 5 |- (x = z -> C = {y | [z / x]y e. C})
14	12, 13	eleq12d 2421	. . . 4 |- (x = z -> (B e. C <-> {y | [z / x]y e. B} e. {y | [z / x]y e. C}))
15	11, 14	sbie 2038	. . 3 |- ([z / x]B e. C <-> {y | [z / x]y e. B} e. {y | [z / x]y e. C})
16	1, 6, 15	vtoclbg 2916	. 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 3138	. . 3 |- [_A / x]_B = {y | [.A / x ].y e. B}
18		df-csb 3138	. . 3 |- [_A / x]_C = {y | [.A / x ].y e. C}
19	17, 18	eleq12i 2418	. 2 |- ([_A / x]_B e. [_A / x]_C <-> {y | [.A / x ].y e. B} e. {y | [.A / x ].y e. C})
20	16, 19	syl6bbr 254	1 |- (A e. V -> ( [.A / x ].B e. C <-> [_A / x]_B e. [_A / x]_C))

Syntax hints:   -> wi 4   <-> wb 176   = wceq 1642  [wsb 1648   e. wcel 1710  {cab 2339   [.wsbc 3047  [_csb 3137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048  df-csb 3138

This theorem is referenced by:  sbcnel12g  3154  sbcel1g  3156  sbcel2g  3158  sbccsb2g  3166