Theorem vtocl2d 29314

Description: Implicit substitution of two classes for two setvar variables. (Contributed by Thierry Arnoux, 25-Aug-2020.)

Hypotheses

Ref	Expression
vtocl2d.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
vtocl2d.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
vtocl2d.1	⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒))
vtocl2d.3	⊢ (𝜑 → 𝜓)

Assertion

Ref	Expression
vtocl2d	⊢ (𝜑 → 𝜒)

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑉   𝑥,𝑊,𝑦   𝜒,𝑥,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝜓(𝑥,𝑦)   𝑉(𝑦)

Proof of Theorem vtocl2d

Step	Hyp	Ref	Expression
1		vtocl2d.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
2		vtocl2d.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		nfcv 2764	. . . 4 ⊢ Ⅎ𝑦𝐵
4		nfcv 2764	. . . 4 ⊢ Ⅎ𝑥𝐵
5		nfcv 2764	. . . 4 ⊢ Ⅎ𝑥𝐴
6		nfv 1843	. . . . 5 ⊢ Ⅎ𝑦𝜑
7		nfsbc1v 3455	. . . . 5 ⊢ Ⅎ𝑦[𝐵 / 𝑦]𝜓
8	6, 7	nfim 1825	. . . 4 ⊢ Ⅎ𝑦(𝜑 → [𝐵 / 𝑦]𝜓)
9		nfv 1843	. . . 4 ⊢ Ⅎ𝑥(𝜑 → 𝜒)
10		sbceq1a 3446	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ [𝐵 / 𝑦]𝜓))
11	10	imbi2d 330	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝜑 → 𝜓) ↔ (𝜑 → [𝐵 / 𝑦]𝜓)))
12		sbceq1a 3446	. . . . . 6 ⊢ (𝑥 = 𝐴 → ([𝐵 / 𝑦]𝜓 ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜓))
13		nfv 1843	. . . . . . . 8 ⊢ Ⅎ𝑥𝜒
14		nfv 1843	. . . . . . . 8 ⊢ Ⅎ𝑦𝜒
15		nfv 1843	. . . . . . . 8 ⊢ Ⅎ𝑥 𝐵 ∈ 𝑊
16		vtocl2d.1	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒))
17	13, 14, 15, 16	sbc2iegf 3504	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓 ↔ 𝜒))
18	2, 1, 17	syl2anc 693	. . . . . 6 ⊢ (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓 ↔ 𝜒))
19	12, 18	sylan9bb 736	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝜑) → ([𝐵 / 𝑦]𝜓 ↔ 𝜒))
20	19	pm5.74da 723	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝜑 → [𝐵 / 𝑦]𝜓) ↔ (𝜑 → 𝜒)))
21		vtocl2d.3	. . . 4 ⊢ (𝜑 → 𝜓)
22	3, 4, 5, 8, 9, 11, 20, 21	vtocl2gf 3268	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝜑 → 𝜒))
23	1, 2, 22	syl2anc 693	. 2 ⊢ (𝜑 → (𝜑 → 𝜒))
24	23	pm2.43i 52	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  [wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sbc 3436

This theorem is referenced by:  submateq  29875