Theorem vtocl2d 28699

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 2751	. . . 4 ⊢ Ⅎ𝑦𝐵
4		nfcv 2751	. . . 4 ⊢ Ⅎ𝑥𝐵
5		nfcv 2751	. . . 4 ⊢ Ⅎ𝑥𝐴
6		nfv 1830	. . . . 5 ⊢ Ⅎ𝑦𝜑
7		nfsbc1v 3422	. . . . 5 ⊢ Ⅎ𝑦[𝐵 / 𝑦]𝜓
8	6, 7	nfim 1813	. . . 4 ⊢ Ⅎ𝑦(𝜑 → [𝐵 / 𝑦]𝜓)
9		nfv 1830	. . . 4 ⊢ Ⅎ𝑥(𝜑 → 𝜒)
10		sbceq1a 3413	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ [𝐵 / 𝑦]𝜓))
11	10	imbi2d 329	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝜑 → 𝜓) ↔ (𝜑 → [𝐵 / 𝑦]𝜓)))
12		sbceq1a 3413	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ([𝐵 / 𝑦]𝜓 ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜓))
13	12	adantr 480	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝜑) → ([𝐵 / 𝑦]𝜓 ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜓))
14		nfv 1830	. . . . . . . . 9 ⊢ Ⅎ𝑥𝜒
15		nfv 1830	. . . . . . . . 9 ⊢ Ⅎ𝑦𝜒
16		nfv 1830	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝐵 ∈ 𝑊
17		vtocl2d.1	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒))
18	14, 15, 16, 17	sbc2iegf 3471	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓 ↔ 𝜒))
19	2, 1, 18	syl2anc 691	. . . . . . 7 ⊢ (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓 ↔ 𝜒))
20	19	adantl 481	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓 ↔ 𝜒))
21	13, 20	bitrd 267	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝜑) → ([𝐵 / 𝑦]𝜓 ↔ 𝜒))
22	21	pm5.74da 719	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝜑 → [𝐵 / 𝑦]𝜓) ↔ (𝜑 → 𝜒)))
23		vtocl2d.3	. . . 4 ⊢ (𝜑 → 𝜓)
24	3, 4, 5, 8, 9, 11, 22, 23	vtocl2gf 3241	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝜑 → 𝜒))
25	1, 2, 24	syl2anc 691	. 2 ⊢ (𝜑 → (𝜑 → 𝜒))
26	25	pm2.43i 50	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  [wsbc 3402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-sbc 3403

This theorem is referenced by:  submateq  29203