Theorem vtoclgaf 2663

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 2229	. . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵
3		vtoclgaf.2	. . . 4 ⊢ Ⅎ𝑥𝜓
4	2, 3	nfim 1504	. . 3 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐵 → 𝜓)
5		eleq1 2141	. . . 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 2657	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐵 → 𝜓))
10	9	pm2.43i 48	1 ⊢ (𝐴 ∈ 𝐵 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 103   = wceq 1284  Ⅎwnf 1389   ∈ wcel 1433  Ⅎwnfc 2206

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-v 2603

This theorem is referenced by:  vtoclga  2664  ssiun2s  3722  tfis  4324  fvmptf  5284  fmptco  5351  prmind2  10502