Theorem vtoclri 3584

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 3207	. 2 ⊢ (𝑥 ∈ 𝐵 → 𝜑)
4	1, 3	vtoclga 3573	1 ⊢ (𝐴 ∈ 𝐵 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1533   ∈ wcel 2110  ∀wral 3138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-nf 1781  df-cleq 2814  df-clel 2893  df-ral 3143

This theorem is referenced by:  alephreg  9998  arch  11888  harmonicbnd  25575  harmonicbnd2  25576  nbgrnself2  27136  heiborlem8  35090  fourierdlem62  42447  srhmsubclem1  44338  srhmsubc  44341  srhmsubcALTV  44359