Theorem sbcralg 2892

Description: Interchange class substitution and restricted quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

Ref	Expression
sbcralg	⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))

Distinct variable groups:   𝑦,𝐴   𝑥,𝐵   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem sbcralg

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 2818	. 2 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ [𝐴 / 𝑥]∀𝑦 ∈ 𝐵 𝜑))
2		dfsbcq2 2818	. . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
3	2	ralbidv 2368	. 2 ⊢ (𝑧 = 𝐴 → (∀𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
4		nfcv 2219	. . . 4 ⊢ Ⅎ𝑥𝐵
5		nfs1v 1856	. . . 4 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
6	4, 5	nfralxy 2402	. . 3 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑
7		sbequ12 1694	. . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
8	7	ralbidv 2368	. . 3 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑))
9	6, 8	sbie 1714	. 2 ⊢ ([𝑧 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑)
10	1, 3, 9	vtoclbg 2659	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))

Syntax hints:   → wi 4   ↔ wb 103   = wceq 1284   ∈ wcel 1433  [wsb 1685  ∀wral 2348  [wsbc 2815

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-sbc 2816

This theorem is referenced by:  r19.12sn  3458