Theorem vtocl2gf 2917

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 25-Apr-1995.)

Hypotheses

Ref	Expression
vtocl2gf.1	⊢ ℲxA
vtocl2gf.2	⊢ ℲyA
vtocl2gf.3	⊢ ℲyB
vtocl2gf.4	⊢ Ⅎxψ
vtocl2gf.5	⊢ Ⅎyχ
vtocl2gf.6	⊢ (x = A → (φ ↔ ψ))
vtocl2gf.7	⊢ (y = B → (ψ ↔ χ))
vtocl2gf.8	⊢ φ

Assertion

Ref	Expression
vtocl2gf	⊢ ((A ∈ V ∧ B ∈ W) → χ)

Proof of Theorem vtocl2gf

Step	Hyp	Ref	Expression
1		elex 2868	. 2 ⊢ (A ∈ V → A ∈ V)
2		vtocl2gf.3	. . 3 ⊢ ℲyB
3		vtocl2gf.2	. . . . 5 ⊢ ℲyA
4	3	nfel1 2500	. . . 4 ⊢ Ⅎy A ∈ V
5		vtocl2gf.5	. . . 4 ⊢ Ⅎyχ
6	4, 5	nfim 1813	. . 3 ⊢ Ⅎy(A ∈ V → χ)
7		vtocl2gf.7	. . . 4 ⊢ (y = B → (ψ ↔ χ))
8	7	imbi2d 307	. . 3 ⊢ (y = B → ((A ∈ V → ψ) ↔ (A ∈ V → χ)))
9		vtocl2gf.1	. . . 4 ⊢ ℲxA
10		vtocl2gf.4	. . . 4 ⊢ Ⅎxψ
11		vtocl2gf.6	. . . 4 ⊢ (x = A → (φ ↔ ψ))
12		vtocl2gf.8	. . . 4 ⊢ φ
13	9, 10, 11, 12	vtoclgf 2914	. . 3 ⊢ (A ∈ V → ψ)
14	2, 6, 8, 13	vtoclgf 2914	. 2 ⊢ (B ∈ W → (A ∈ V → χ))
15	1, 14	mpan9 455	1 ⊢ ((A ∈ V ∧ B ∈ W) → χ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  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-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:  vtocl3gf  2918  vtocl2g  2919  vtocl2gaf  2922