Theorem vtocl2g 3531

Description: Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 25-Apr-1995.) Remove dependency on ax-10 2147, ax-11 2163, and ax-13 2377. (Revised by Steven Nguyen, 29-Nov-2022.)

Hypotheses

Ref	Expression
vtocl2g.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
vtocl2g.2	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
vtocl2g.3	⊢ 𝜑

Assertion

Ref	Expression
vtocl2g	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝜒)

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

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)   𝜒(𝑥)   𝐵(𝑥)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem vtocl2g

Step	Hyp	Ref	Expression
1		elex 3463	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		vtocl2g.2	. . . 4 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
3	2	imbi2d 340	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ V → 𝜓) ↔ (𝐴 ∈ V → 𝜒)))
4		vtocl2g.1	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
5		vtocl2g.3	. . . 4 ⊢ 𝜑
6	4, 5	vtoclg 3513	. . 3 ⊢ (𝐴 ∈ V → 𝜓)
7	3, 6	vtoclg 3513	. 2 ⊢ (𝐵 ∈ 𝑊 → (𝐴 ∈ V → 𝜒))
8	1, 7	mpan9 506	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝜒)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3442

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-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444

This theorem is referenced by:  vtocl3g  3532  vtocl4g  3542  opthg  5435  opelopabsb  5488  vtoclr  5697  funopg  6536  f1osng  6826  fsng  7094  fnpr2g  7168  unexbOLD  7705  op1stg  7957  op2ndg  7958  xpsneng  9004  xpcomeng  9011  sbth  9039  sbthfi  9137  unxpdom  9173  prcdnq  10918  mhmlem  19009  carsgmon  34498  brimageg  36147  brdomaing  36155  brrangeg  36156  rankung  36388  mbfresfi  37946  zindbi  43332  2sbc6g  44800  2sbc5g  44801  fmulcl  45970