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  3903  intprg  3956  opthg  4324  opelopabsb  4348  unexb  4533  vtoclr  4767  elimasng  5096  cnvsng  5214  funopg  5352  f1osng  5616  fsng  5810  fvsng  5839  op1stg  6302  op2ndg  6303  xpsneng  6989  xpcomeng  6995  mhmlem  13666  bdunexb  16338  bj-unexg  16339