Theorem vtocl2ga 2754

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 2281	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2281	. 2 ⊢ Ⅎ𝑦𝐴
3		nfcv 2281	. 2 ⊢ Ⅎ𝑦𝐵
4		nfv 1508	. 2 ⊢ Ⅎ𝑥𝜓
5		nfv 1508	. 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 2753	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331   ∈ wcel 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688

This theorem is referenced by:  caovcan  5935  genipv  7317  fsumrelem  11240