Theorem vtoclgaf 2795

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 17-Feb-2006.) (Revised by Mario Carneiro, 10-Oct-2016.)

Hypotheses

Ref	Expression
vtoclgaf.1	⊢ Ⅎ𝑥𝐴
vtoclgaf.2	⊢ Ⅎ𝑥𝜓
vtoclgaf.3	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
vtoclgaf.4	⊢ (𝑥 ∈ 𝐵 → 𝜑)

Assertion

Ref	Expression
vtoclgaf	⊢ (𝐴 ∈ 𝐵 → 𝜓)

Distinct variable group:   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem vtoclgaf

Step	Hyp	Ref	Expression
1		vtoclgaf.1	. . 3 ⊢ Ⅎ𝑥𝐴
2	1	nfel1 2323	. . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵
3		vtoclgaf.2	. . . 4 ⊢ Ⅎ𝑥𝜓
4	2, 3	nfim 1565	. . 3 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐵 → 𝜓)
5		eleq1 2233	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
6		vtoclgaf.3	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
7	5, 6	imbi12d 233	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 → 𝜑) ↔ (𝐴 ∈ 𝐵 → 𝜓)))
8		vtoclgaf.4	. . 3 ⊢ (𝑥 ∈ 𝐵 → 𝜑)
9	1, 4, 7, 8	vtoclgf 2788	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐵 → 𝜓))
10	9	pm2.43i 49	1 ⊢ (𝐴 ∈ 𝐵 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1348  Ⅎwnf 1453   ∈ wcel 2141  Ⅎwnfc 2299

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

This theorem is referenced by:  vtoclga  2796  ssiun2s  3917  tfis  4567  fvmptf  5588  fmptco  5662  prmind2  12074