Theorem vtocl2gf 2717

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 2666	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		vtocl2gf.3	. . 3 ⊢ Ⅎ𝑦𝐵
3		vtocl2gf.2	. . . . 5 ⊢ Ⅎ𝑦𝐴
4	3	nfel1 2264	. . . 4 ⊢ Ⅎ𝑦 𝐴 ∈ V
5		vtocl2gf.5	. . . 4 ⊢ Ⅎ𝑦𝜒
6	4, 5	nfim 1532	. . 3 ⊢ Ⅎ𝑦(𝐴 ∈ V → 𝜒)
7		vtocl2gf.7	. . . 4 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
8	7	imbi2d 229	. . 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 2713	. . 3 ⊢ (𝐴 ∈ V → 𝜓)
14	2, 6, 8, 13	vtoclgf 2713	. 2 ⊢ (𝐵 ∈ 𝑊 → (𝐴 ∈ V → 𝜒))
15	1, 14	mpan9 277	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1312  Ⅎwnf 1417   ∈ wcel 1461  Ⅎwnfc 2240  Vcvv 2655

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657

This theorem is referenced by:  vtocl3gf  2718  vtocl2g  2719  vtocl2gaf  2722