Theorem vtoclgaf 3424

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 1995	. . 3 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐵 → 𝜓)
5		eleq1 2832	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
6		vtoclgaf.3	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
7	5, 6	imbi12d 335	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 → 𝜑) ↔ (𝐴 ∈ 𝐵 → 𝜓)))
8		vtoclgaf.4	. . 3 ⊢ (𝑥 ∈ 𝐵 → 𝜑)
9	1, 4, 7, 8	vtoclgf 3416	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐵 → 𝜓))
10	9	pm2.43i 52	1 ⊢ (𝐴 ∈ 𝐵 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 197   = wceq 1652  Ⅎwnf 1878   ∈ wcel 2155  Ⅎwnfc 2894

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352

This theorem is referenced by:  vtoclga  3425  ssiun2s  4722  iunopeqop  5144  fvmptss  6483  fvmptf  6492  fmptco  6589  tfis  7254  inar1  9852  sumss  14743  fprodn0  14995  prmind2  15681  lss1d  19238  itg2splitlem  23809  dgrle  24293  cnlnadjlem5  29389  poimirlem25  33861  stoweidlem26  40883