Theorem vtoclgaf 2677

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 2235	. . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵
3		vtoclgaf.2	. . . 4 ⊢ Ⅎ𝑥𝜓
4	2, 3	nfim 1507	. . 3 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐵 → 𝜓)
5		eleq1 2147	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
6		vtoclgaf.3	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
7	5, 6	imbi12d 232	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 → 𝜑) ↔ (𝐴 ∈ 𝐵 → 𝜓)))
8		vtoclgaf.4	. . 3 ⊢ (𝑥 ∈ 𝐵 → 𝜑)
9	1, 4, 7, 8	vtoclgf 2671	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐵 → 𝜓))
10	9	pm2.43i 48	1 ⊢ (𝐴 ∈ 𝐵 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 103   = wceq 1287  Ⅎwnf 1392   ∈ wcel 1436  Ⅎwnfc 2212

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 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067

This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617

This theorem is referenced by:  vtoclga  2678  ssiun2s  3759  tfis  4373  fvmptf  5360  fmptco  5429  prmind2  11008