Theorem vtocl2gaf 2831

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 2350	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐶
5		nfv 1542	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐷
6	4, 5	nfan 1579	. . . 4 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)
7		vtocl2gaf.1	. . . 4 ⊢ Ⅎ𝑥𝜓
8	6, 7	nfim 1586	. . 3 ⊢ Ⅎ𝑥((𝐴 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → 𝜓)
9	2	nfel1 2350	. . . . 5 ⊢ Ⅎ𝑦 𝐴 ∈ 𝐶
10	3	nfel1 2350	. . . . 5 ⊢ Ⅎ𝑦 𝐵 ∈ 𝐷
11	9, 10	nfan 1579	. . . 4 ⊢ Ⅎ𝑦(𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)
12		vtocl2gaf.2	. . . 4 ⊢ Ⅎ𝑦𝜒
13	11, 12	nfim 1586	. . 3 ⊢ Ⅎ𝑦((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝜒)
14		eleq1 2259	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶))
15	14	anbi1d 465	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ (𝐴 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)))
16		vtocl2gaf.3	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
17	15, 16	imbi12d 234	. . 3 ⊢ (𝑥 = 𝐴 → (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → 𝜑) ↔ ((𝐴 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → 𝜓)))
18		eleq1 2259	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝐷 ↔ 𝐵 ∈ 𝐷))
19	18	anbi2d 464	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)))
20		vtocl2gaf.4	. . . 4 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
21	19, 20	imbi12d 234	. . 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 2826	. 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 104   ↔ wb 105   = wceq 1364  Ⅎwnf 1474   ∈ wcel 2167  Ⅎwnfc 2326

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765

This theorem is referenced by:  vtocl2ga  2832  ovmpos  6046  ov2gf  6047  ovi3  6060  cnmptcom  14534