Theorem vtocl2gaf 3545

Description: Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 10-Aug-2013.) (Proof shortened by Wolf Lammen, 31-May-2025.)

Hypotheses

Ref	Expression
vtocl2gaf.a	⊢ Ⅎ𝑥𝐴
vtocl2gaf.b	⊢ Ⅎ𝑦𝐴
vtocl2gaf.c	⊢ Ⅎ𝑦𝐵
vtocl2gaf.1	⊢ Ⅎ𝑥𝜓
vtocl2gaf.2	⊢ Ⅎ𝑦𝜒
vtocl2gaf.3	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
vtocl2gaf.4	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
vtocl2gaf.5	⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → 𝜑)

Assertion

Ref	Expression
vtocl2gaf	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝜒)

Distinct variable groups:   𝑥,𝑦,𝐶   𝑥,𝐷,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝜓(𝑥, 𝑦)   𝜒(𝑥, 𝑦)   𝐴(𝑥, 𝑦)   𝐵(𝑥, 𝑦)

Proof of Theorem vtocl2gaf

Step	Hyp	Ref	Expression
1		vtocl2gaf.c	. . 3 ⊢ Ⅎ𝑦𝐵
2		vtocl2gaf.b	. . . . 5 ⊢ Ⅎ𝑦𝐴
3	2	nfel1 2943	. . . 4 ⊢ Ⅎ𝑦 𝐴 ∈ 𝐶
4		vtocl2gaf.2	. . . 4 ⊢ Ⅎ𝑦𝜒
5	3, 4	nfim 1929	. . 3 ⊢ Ⅎ𝑦(𝐴 ∈ 𝐶 → 𝜒)
6		vtocl2gaf.4	. . . 4 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
7	6	imbi2d 343	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ 𝐶 → 𝜓) ↔ (𝐴 ∈ 𝐶 → 𝜒)))
8		vtocl2gaf.a	. . . . 5 ⊢ Ⅎ𝑥𝐴
9		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐷
10		vtocl2gaf.1	. . . . . 6 ⊢ Ⅎ𝑥𝜓
11	9, 10	nfim 1929	. . . . 5 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐷 → 𝜓)
12		vtocl2gaf.3	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
13	12	imbi2d 343	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑦 ∈ 𝐷 → 𝜑) ↔ (𝑦 ∈ 𝐷 → 𝜓)))
14		vtocl2gaf.5	. . . . . 6 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → 𝜑)
15	14	ex 418	. . . . 5 ⊢ (𝑥 ∈ 𝐶 → (𝑦 ∈ 𝐷 → 𝜑))
16	8, 11, 13, 15	vtoclgaf 3542	. . . 4 ⊢ (𝐴 ∈ 𝐶 → (𝑦 ∈ 𝐷 → 𝜓))
17	16	com12 33	. . 3 ⊢ (𝑦 ∈ 𝐷 → (𝐴 ∈ 𝐶 → 𝜓))
18	1, 5, 7, 17	vtoclgaf 3542	. 2 ⊢ (𝐵 ∈ 𝐷 → (𝐴 ∈ 𝐶 → 𝜒))
19	18	impcom 413	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝜒)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2146  Ⅎwnfc 2912

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-v 3459

This theorem is used by:  vtocl3gaf  3546  ovmpos  7567  ov2gf  7568  ov3  7582  pwfseqlem2  10661  cnmptcom  23888