Theorem vtocl2g 3558

Description: Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 25-Apr-1995.) Remove dependency on ax-10 2142, ax-11 2158, 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 3485	. 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 3538	. . 3 ⊢ (𝐴 ∈ V → 𝜓)
7	3, 6	vtoclg 3538	. 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 1540   ∈ wcel 2109  Vcvv 3464

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-v 3466

This theorem is referenced by:  vtocl3g  3559  vtocl4g  3569  opthg  5457  opelopabsb  5510  vtoclr  5722  funopg  6575  f1osng  6864  fsng  7132  fnpr2g  7207  unexbOLD  7747  op1stg  8005  op2ndg  8006  xpsneng  9075  xpcomeng  9083  sbth  9112  sbthfi  9218  unxpdom  9266  prcdnq  11012  mhmlem  19050  carsgmon  34351  brimageg  35950  brdomaing  35958  brrangeg  35959  rankung  36189  mbfresfi  37695  zindbi  42937  2sbc6g  44406  2sbc5g  44407  fmulcl  45577