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

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   = wceq 1642  [wsb 1648   ∈ wcel 1710  {cab 2339  [̣wsbc 3047  [csb 3137

This proof depends on 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 proof 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 used by:  sbcnel12g  3154  sbcel1g  3156  sbcel2g  3158  sbccsb2g  3166