Theorem vtoclgaf 3576

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 2922	. . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵
3		vtoclgaf.2	. . . 4 ⊢ Ⅎ𝑥𝜓
4	2, 3	nfim 1896	. . 3 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐵 → 𝜓)
5		eleq1 2829	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
6		vtoclgaf.3	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
7	5, 6	imbi12d 344	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 → 𝜑) ↔ (𝐴 ∈ 𝐵 → 𝜓)))
8		vtoclgaf.4	. . 3 ⊢ (𝑥 ∈ 𝐵 → 𝜑)
9	1, 4, 7, 8	vtoclgf 3569	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐵 → 𝜓))
10	9	pm2.43i 52	1 ⊢ (𝐴 ∈ 𝐵 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2108  Ⅎwnfc 2890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-v 3482

This theorem is referenced by:  vtocl2gaf  3579  vtocl3gaf  3581  ssiun2s  5048  iunopeqop  5526  fvmptss  7028  fvmptf  7037  fmptco  7149  tfis  7876  inar1  10815  sumss  15760  fprodn0  16015  prmind2  16722  lss1d  20961  itg2splitlem  25783  dgrle  26282  cnlnadjlem5  32090  poimirlem25  37652  stoweidlem26  46041