Theorem vtocl3gf 3535

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 3474	. . 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 2940	. . . . 5 ⊢ Ⅎ𝑦 𝐴 ∈ V
7		vtocl3gf.2	. . . . 5 ⊢ Ⅎ𝑦𝜒
8	6, 7	nfim 1929	. . . 4 ⊢ Ⅎ𝑦(𝐴 ∈ V → 𝜒)
9		vtocl3gf.c	. . . . . 6 ⊢ Ⅎ𝑧𝐴
10	9	nfel1 2940	. . . . 5 ⊢ Ⅎ𝑧 𝐴 ∈ V
11		vtocl3gf.3	. . . . 5 ⊢ Ⅎ𝑧𝜃
12	10, 11	nfim 1929	. . . 4 ⊢ Ⅎ𝑧(𝐴 ∈ V → 𝜃)
13		vtocl3gf.5	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
14	13	imbi2d 343	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ V → 𝜓) ↔ (𝐴 ∈ V → 𝜒)))
15		vtocl3gf.6	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))
16	15	imbi2d 343	. . . 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 3532	. . . 4 ⊢ (𝐴 ∈ V → 𝜓)
22	2, 3, 4, 8, 12, 14, 16, 21	vtocl2gf 3534	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 ∈ V → 𝜃))
23	1, 22	mpan9 516	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋)) → 𝜃)
24	23	3impb 1132	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝜃)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2145  Ⅎwnfc 2909  Vcvv 3453

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-v 3455

This theorem is used by: (None)