Theorem vtocl2g 3533

Description: Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 25-Apr-1995.) Remove dependency on ax-10 2177, ax-11 2193, and ax-13 2401. (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 3471	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		vtocl2g.2	. . . 4 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
3	2	imbi2d 343	. . 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 3517	. . 3 ⊢ (𝐴 ∈ V → 𝜓)
7	3, 6	vtoclg 3517	. 2 ⊢ (𝐵 ∈ 𝑊 → (𝐴 ∈ V → 𝜒))
8	1, 7	mpan9 516	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝜒)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Vcvv 3450

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452

This theorem is used by:  vtocl3g  3534  vtocl4g  3541  opthg  5446  opelopabsb  5504  vtoclr  5714  funopg  6579  f1osng  6872  fsng  7143  fnpr2g  7220  op1stg  8017  op2ndg  8018  xpsneng  9088  xpcomeng  9095  sbth  9123  sbthfi  9221  unxpdom  9257  rankung  9880  prcdnq  11085  mhmlem  19281  carsgmon  34996  brimageg  36785  brdomaing  36793  brrangeg  36794  mbfresfi  38693  zindbi  44062  2sbc6g  45508  2sbc5g  45509  fmulcl  46701