Theorem vtocl2ga 2666

Description: Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 20-Aug-1995.)

Hypotheses

Ref	Expression
vtocl2ga.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
vtocl2ga.2	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
vtocl2ga.3	⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → 𝜑)

Assertion

Ref	Expression
vtocl2ga	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝜒)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦   𝜓,𝑥   𝜒,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)   𝜒(𝑥)   𝐵(𝑥)

Proof of Theorem vtocl2ga

Step	Hyp	Ref	Expression
1		nfcv 2219	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2219	. 2 ⊢ Ⅎ𝑦𝐴
3		nfcv 2219	. 2 ⊢ Ⅎ𝑦𝐵
4		nfv 1461	. 2 ⊢ Ⅎ𝑥𝜓
5		nfv 1461	. 2 ⊢ Ⅎ𝑦𝜒
6		vtocl2ga.1	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
7		vtocl2ga.2	. 2 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
8		vtocl2ga.3	. 2 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → 𝜑)
9	1, 2, 3, 4, 5, 6, 7, 8	vtocl2gaf 2665	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284   ∈ wcel 1433

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603

This theorem is referenced by:  caovcan  5685  genipv  6699