Theorem vtocl3gf 2827

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)

Hypotheses

Ref	Expression
vtocl3gf.a	⊢ Ⅎ𝑥𝐴
vtocl3gf.b	⊢ Ⅎ𝑦𝐴
vtocl3gf.c	⊢ Ⅎ𝑧𝐴
vtocl3gf.d	⊢ Ⅎ𝑦𝐵
vtocl3gf.e	⊢ Ⅎ𝑧𝐵
vtocl3gf.f	⊢ Ⅎ𝑧𝐶
vtocl3gf.1	⊢ Ⅎ𝑥𝜓
vtocl3gf.2	⊢ Ⅎ𝑦𝜒
vtocl3gf.3	⊢ Ⅎ𝑧𝜃
vtocl3gf.4	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
vtocl3gf.5	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
vtocl3gf.6	⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))
vtocl3gf.7	⊢ 𝜑

Assertion

Ref	Expression
vtocl3gf	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝜃)

Proof of Theorem vtocl3gf

Step	Hyp	Ref	Expression
1		elex 2774	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		vtocl3gf.d	. . . 4 ⊢ Ⅎ𝑦𝐵
3		vtocl3gf.e	. . . 4 ⊢ Ⅎ𝑧𝐵
4		vtocl3gf.f	. . . 4 ⊢ Ⅎ𝑧𝐶
5		vtocl3gf.b	. . . . . 6 ⊢ Ⅎ𝑦𝐴
6	5	nfel1 2350	. . . . 5 ⊢ Ⅎ𝑦 𝐴 ∈ V
7		vtocl3gf.2	. . . . 5 ⊢ Ⅎ𝑦𝜒
8	6, 7	nfim 1586	. . . 4 ⊢ Ⅎ𝑦(𝐴 ∈ V → 𝜒)
9		vtocl3gf.c	. . . . . 6 ⊢ Ⅎ𝑧𝐴
10	9	nfel1 2350	. . . . 5 ⊢ Ⅎ𝑧 𝐴 ∈ V
11		vtocl3gf.3	. . . . 5 ⊢ Ⅎ𝑧𝜃
12	10, 11	nfim 1586	. . . 4 ⊢ Ⅎ𝑧(𝐴 ∈ V → 𝜃)
13		vtocl3gf.5	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
14	13	imbi2d 230	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ V → 𝜓) ↔ (𝐴 ∈ V → 𝜒)))
15		vtocl3gf.6	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))
16	15	imbi2d 230	. . . 4 ⊢ (𝑧 = 𝐶 → ((𝐴 ∈ V → 𝜒) ↔ (𝐴 ∈ V → 𝜃)))
17		vtocl3gf.a	. . . . 5 ⊢ Ⅎ𝑥𝐴
18		vtocl3gf.1	. . . . 5 ⊢ Ⅎ𝑥𝜓
19		vtocl3gf.4	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
20		vtocl3gf.7	. . . . 5 ⊢ 𝜑
21	17, 18, 19, 20	vtoclgf 2822	. . . 4 ⊢ (𝐴 ∈ V → 𝜓)
22	2, 3, 4, 8, 12, 14, 16, 21	vtocl2gf 2826	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 ∈ V → 𝜃))
23	1, 22	mpan9 281	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋)) → 𝜃)
24	23	3impb 1201	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝜃)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364  Ⅎwnf 1474   ∈ wcel 2167  Ⅎwnfc 2326  Vcvv 2763

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-3an 982  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:  vtocl3gaf  2833