Theorem vtocl2 3476

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,𝑦   𝑥,𝐴   𝑥,𝐵,𝑦   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		vtocl2.4	. . . 4 ⊢ 𝜑
3	2	a1i 11	. . 3 ⊢ (𝑦 = 𝐵 → 𝜑)
4		vtocl2.1	. . . 4 ⊢ 𝐴 ∈ V
5		vtocl2.3	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))
6	5	pm5.74da 804	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑦 = 𝐵 → 𝜑) ↔ (𝑦 = 𝐵 → 𝜓)))
7	4, 6, 3	vtocl 3474	. . 3 ⊢ (𝑦 = 𝐵 → 𝜓)
8	3, 7	2thd 268	. 2 ⊢ (𝑦 = 𝐵 → (𝜑 ↔ 𝜓))
9	1, 8, 2	vtocl 3474	1 ⊢ 𝜓

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2110  Vcvv 3408

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-clel 2816

This theorem is referenced by:  vtocl3  3477  caovord  7419  sornom  9891  wloglei  11364  ipodrsima  18047  mpfind  21067  mclsppslem  33258  monotoddzzfi  40467