Theorem vtocl2gaf 2748

Description: Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 10-Aug-2013.)

Hypotheses

Ref	Expression
vtocl2gaf.a	⊢ Ⅎ𝑥𝐴
vtocl2gaf.b	⊢ Ⅎ𝑦𝐴
vtocl2gaf.c	⊢ Ⅎ𝑦𝐵
vtocl2gaf.1	⊢ Ⅎ𝑥𝜓
vtocl2gaf.2	⊢ Ⅎ𝑦𝜒
vtocl2gaf.3	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
vtocl2gaf.4	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
vtocl2gaf.5	⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → 𝜑)

Assertion

Ref	Expression
vtocl2gaf	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝜒)

Distinct variable groups:   𝑥,𝑦,𝐶   𝑥,𝐷,𝑦

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

Proof of Theorem vtocl2gaf

Step	Hyp	Ref	Expression
1		vtocl2gaf.a	. . 3 ⊢ Ⅎ𝑥𝐴
2		vtocl2gaf.b	. . 3 ⊢ Ⅎ𝑦𝐴
3		vtocl2gaf.c	. . 3 ⊢ Ⅎ𝑦𝐵
4	1	nfel1 2290	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐶
5		nfv 1508	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐷
6	4, 5	nfan 1544	. . . 4 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)
7		vtocl2gaf.1	. . . 4 ⊢ Ⅎ𝑥𝜓
8	6, 7	nfim 1551	. . 3 ⊢ Ⅎ𝑥((𝐴 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → 𝜓)
9	2	nfel1 2290	. . . . 5 ⊢ Ⅎ𝑦 𝐴 ∈ 𝐶
10	3	nfel1 2290	. . . . 5 ⊢ Ⅎ𝑦 𝐵 ∈ 𝐷
11	9, 10	nfan 1544	. . . 4 ⊢ Ⅎ𝑦(𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)
12		vtocl2gaf.2	. . . 4 ⊢ Ⅎ𝑦𝜒
13	11, 12	nfim 1551	. . 3 ⊢ Ⅎ𝑦((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝜒)
14		eleq1 2200	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶))
15	14	anbi1d 460	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ (𝐴 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)))
16		vtocl2gaf.3	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
17	15, 16	imbi12d 233	. . 3 ⊢ (𝑥 = 𝐴 → (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → 𝜑) ↔ ((𝐴 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → 𝜓)))
18		eleq1 2200	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝐷 ↔ 𝐵 ∈ 𝐷))
19	18	anbi2d 459	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)))
20		vtocl2gaf.4	. . . 4 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
21	19, 20	imbi12d 233	. . 3 ⊢ (𝑦 = 𝐵 → (((𝐴 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → 𝜓) ↔ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝜒)))
22		vtocl2gaf.5	. . 3 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → 𝜑)
23	1, 2, 3, 8, 13, 17, 21, 22	vtocl2gf 2743	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝜒))
24	23	pm2.43i 49	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331  Ⅎwnf 1436   ∈ wcel 1480  Ⅎwnfc 2266

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 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683

This theorem is referenced by:  vtocl2ga  2749  ovmpos  5887  ov2gf  5888  ovi3  5900  cnmptcom  12456