Theorem vtocl2 3540

Description: Implicit substitution of classes for setvar variables. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Hypotheses

Ref	Expression
vtocl2.1	⊢ 𝐴 ∈ V
vtocl2.2	⊢ 𝐵 ∈ V
vtocl2.3	⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))
vtocl2.4	⊢ 𝜑

Assertion

Ref	Expression
vtocl2	⊢ 𝜓

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

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑦)

Proof of Theorem vtocl2

Step	Hyp	Ref	Expression
1		vtocl2.2	. 2 ⊢ 𝐵 ∈ V
2		vtocl2.1	. . 3 ⊢ 𝐴 ∈ V
3		vtocl2.3	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))
4	3	pm5.74da 815	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑦 = 𝐵 → 𝜑) ↔ (𝑦 = 𝐵 → 𝜓)))
5		vtocl2.4	. . . 4 ⊢ 𝜑
6	5	a1i 11	. . 3 ⊢ (𝑦 = 𝐵 → 𝜑)
7	2, 4, 6	vtocl 3534	. 2 ⊢ (𝑦 = 𝐵 → 𝜓)
8	1, 7	vtocle 3532	1 ⊢ 𝜓

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-clel 2844

This theorem is referenced by:  vtocl3  3541  caovord  7622  sornom  10261  wloglei  11746  ipodrsima  18597  mpfind  22235  mclsppslem  36008  mh-inf3f1  36975  monotoddzzfi  43595