Theorem vtocl2g 2836

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 2347	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2347	. 2 ⊢ Ⅎ𝑦𝐴
3		nfcv 2347	. 2 ⊢ Ⅎ𝑦𝐵
4		nfv 1550	. 2 ⊢ Ⅎ𝑥𝜓
5		nfv 1550	. 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 2834	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1372   ∈ wcel 2175

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773

This theorem is referenced by:  uniprg  3864  intprg  3917  opthg  4281  opelopabsb  4305  unexb  4488  vtoclr  4722  elimasng  5049  cnvsng  5167  funopg  5304  f1osng  5562  fsng  5752  fvsng  5779  op1stg  6235  op2ndg  6236  xpsneng  6916  xpcomeng  6922  mhmlem  13392  bdunexb  15789  bj-unexg  15790