Theorem vtoclri 3532

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 3228	. 2 ⊢ (𝑥 ∈ 𝐵 → 𝜑)
4	1, 3	vtoclga 3520	1 ⊢ (𝐴 ∈ 𝐵 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542   ∈ wcel 2114  ∀wral 3051

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-12 2185  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052

This theorem is referenced by:  alephreg  10505  arch  12434  srhmsubclem1  20654  srhmsubc  20657  harmonicbnd  26967  harmonicbnd2  26968  nbgrnself2  29429  heiborlem8  38139  fourierdlem62  46596  natglobalincr  47307  srhmsubcALTV  48801