Theorem vtocl2gf 3570

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 3513	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		vtocl2gf.3	. . 3 ⊢ Ⅎ𝑦𝐵
3		vtocl2gf.2	. . . . 5 ⊢ Ⅎ𝑦𝐴
4	3	nfel1 2994	. . . 4 ⊢ Ⅎ𝑦 𝐴 ∈ V
5		vtocl2gf.5	. . . 4 ⊢ Ⅎ𝑦𝜒
6	4, 5	nfim 1893	. . 3 ⊢ Ⅎ𝑦(𝐴 ∈ V → 𝜒)
7		vtocl2gf.7	. . . 4 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
8	7	imbi2d 343	. . 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 3566	. . 3 ⊢ (𝐴 ∈ V → 𝜓)
14	2, 6, 8, 13	vtoclgf 3566	. 2 ⊢ (𝐵 ∈ 𝑊 → (𝐴 ∈ V → 𝜒))
15	1, 14	mpan9 509	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝜒)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533  Ⅎwnf 1780   ∈ wcel 2110  Ⅎwnfc 2961  Vcvv 3495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3497

This theorem is referenced by:  vtocl3gf  3571  vtocl2gaf  3576  vtocl2dOLD  3931  offval22  7777  fmuldfeqlem1  41855