Theorem vtocl2g 2865

Description: Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 25-Apr-1995.)

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		nfcv 2372	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2372	. 2 ⊢ Ⅎ𝑦𝐴
3		nfcv 2372	. 2 ⊢ Ⅎ𝑦𝐵
4		nfv 1574	. 2 ⊢ Ⅎ𝑥𝜓
5		nfv 1574	. 2 ⊢ Ⅎ𝑦𝜒
6		vtocl2g.1	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
7		vtocl2g.2	. 2 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
8		vtocl2g.3	. 2 ⊢ 𝜑
9	1, 2, 3, 4, 5, 6, 7, 8	vtocl2gf 2863	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801

This theorem is referenced by:  vtocl4g  2872  uniprg  3902  intprg  3955  opthg  4323  opelopabsb  4347  unexb  4532  vtoclr  4766  elimasng  5095  cnvsng  5213  funopg  5351  f1osng  5613  fsng  5807  fvsng  5834  op1stg  6294  op2ndg  6295  xpsneng  6977  xpcomeng  6983  mhmlem  13646  bdunexb  16241  bj-unexg  16242