Theorem vtocl2gf 2826

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 2774	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		vtocl2gf.3	. . 3 ⊢ Ⅎ𝑦𝐵
3		vtocl2gf.2	. . . . 5 ⊢ Ⅎ𝑦𝐴
4	3	nfel1 2350	. . . 4 ⊢ Ⅎ𝑦 𝐴 ∈ V
5		vtocl2gf.5	. . . 4 ⊢ Ⅎ𝑦𝜒
6	4, 5	nfim 1586	. . 3 ⊢ Ⅎ𝑦(𝐴 ∈ V → 𝜒)
7		vtocl2gf.7	. . . 4 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
8	7	imbi2d 230	. . 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 2822	. . 3 ⊢ (𝐴 ∈ V → 𝜓)
14	2, 6, 8, 13	vtoclgf 2822	. 2 ⊢ (𝐵 ∈ 𝑊 → (𝐴 ∈ V → 𝜒))
15	1, 14	mpan9 281	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364  Ⅎwnf 1474   ∈ wcel 2167  Ⅎwnfc 2326  Vcvv 2763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765

This theorem is referenced by:  vtocl3gf  2827  vtocl2g  2828  vtocl2gaf  2831