Theorem vtocl2g 3500

Description: Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 25-Apr-1995.) Remove dependency on ax-10 2139, ax-11 2156, and ax-13 2372. (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 3440	. 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 3495	. . 3 ⊢ (𝐴 ∈ V → 𝜓)
7	3, 6	vtoclg 3495	. 2 ⊢ (𝐵 ∈ 𝑊 → (𝐴 ∈ V → 𝜒))
8	1, 7	mpan9 506	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝜒)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Vcvv 3422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424

This theorem is referenced by:  vtocl3g  3501  vtocl4g  3509  uniprgOLD  4856  intprgOLD  4912  opthg  5386  opelopabsb  5436  vtoclr  5641  elimasngOLD  5987  funopg  6452  f1osng  6740  fsng  6991  fnpr2g  7068  unexb  7576  op1stg  7816  op2ndg  7817  xpsneng  8797  xpcomeng  8804  sbth  8833  sbthfi  8942  unxpdom  8959  fpwwe2lem4  10321  prcdnq  10680  mhmlem  18610  carsgmon  32181  brimageg  34156  brdomaing  34164  brrangeg  34165  rankung  34395  mbfresfi  35750  zindbi  40684  2sbc6g  41922  2sbc5g  41923  fmulcl  43012