Theorem vtoclri 3573

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 21-Nov-1994.)

Hypotheses

Ref	Expression
vtoclri.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
vtoclri.2	⊢ ∀𝑥 ∈ 𝐵 𝜑

Assertion

Ref	Expression
vtoclri	⊢ (𝐴 ∈ 𝐵 → 𝜓)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem vtoclri

Step	Hyp	Ref	Expression
1		vtoclri.1	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
2		vtoclri.2	. . 3 ⊢ ∀𝑥 ∈ 𝐵 𝜑
3	2	rspec 3246	. 2 ⊢ (𝑥 ∈ 𝐵 → 𝜑)
4	1, 3	vtoclga 3562	1 ⊢ (𝐴 ∈ 𝐵 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1541   ∈ wcel 2106  ∀wral 3060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-12 2171  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061

This theorem is referenced by:  alephreg  10559  arch  12451  harmonicbnd  26435  harmonicbnd2  26436  nbgrnself2  28482  heiborlem8  36491  fourierdlem62  44657  natglobalincr  45364  srhmsubclem1  46619  srhmsubc  46622  srhmsubcALTV  46640