Theorem vtocl3gf 2918

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 2868	. . 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 2500	. . . . 5 ⊢ Ⅎy A ∈ V
7		vtocl3gf.2	. . . . 5 ⊢ Ⅎyχ
8	6, 7	nfim 1813	. . . 4 ⊢ Ⅎy(A ∈ V → χ)
9		vtocl3gf.c	. . . . . 6 ⊢ ℲzA
10	9	nfel1 2500	. . . . 5 ⊢ Ⅎz A ∈ V
11		vtocl3gf.3	. . . . 5 ⊢ Ⅎzθ
12	10, 11	nfim 1813	. . . 4 ⊢ Ⅎz(A ∈ V → θ)
13		vtocl3gf.5	. . . . 5 ⊢ (y = B → (ψ ↔ χ))
14	13	imbi2d 307	. . . 4 ⊢ (y = B → ((A ∈ V → ψ) ↔ (A ∈ V → χ)))
15		vtocl3gf.6	. . . . 5 ⊢ (z = C → (χ ↔ θ))
16	15	imbi2d 307	. . . 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 2914	. . . 4 ⊢ (A ∈ V → ψ)
22	2, 3, 4, 8, 12, 14, 16, 21	vtocl2gf 2917	. . 3 ⊢ ((B ∈ W ∧ C ∈ X) → (A ∈ V → θ))
23	1, 22	mpan9 455	. 2 ⊢ ((A ∈ V ∧ (B ∈ W ∧ C ∈ X)) → θ)
24	23	3impb 1147	1 ⊢ ((A ∈ V ∧ B ∈ W ∧ C ∈ X) → θ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2477  Vcvv 2860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862

This theorem is referenced by:  vtocl3gaf  2924