Theorem vtocl2d 3544

Description: Implicit substitution of two classes for two setvar variables. (Contributed by Thierry Arnoux, 25-Aug-2020.) (Revised by BTernaryTau, 19-Oct-2023.)

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   𝑥,𝐵,𝑦   𝜒,𝑥,𝑦   𝜑,𝑥,𝑦

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

Proof of Theorem vtocl2d

Step	Hyp	Ref	Expression
1		vtocl2d.b	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
2		vtocl2d.3	. . . 4 ⊢ (𝜑 → 𝜓)
3	2	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → 𝜓)
4		vtocl2d.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5		vtocl2d.1	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒))
6	5	adantll 711	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒))
7	6	pm5.74da 801	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝑦 = 𝐵 → 𝜓) ↔ (𝑦 = 𝐵 → 𝜒)))
8	2	a1d 25	. . . . 5 ⊢ (𝜑 → (𝑦 = 𝐵 → 𝜓))
9	4, 7, 8	vtocld 3542	. . . 4 ⊢ (𝜑 → (𝑦 = 𝐵 → 𝜒))
10	9	imp 406	. . 3 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → 𝜒)
11	3, 10	2thd 265	. 2 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒))
12	1, 11, 2	vtocld 3542	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-clel 2804

This theorem is referenced by:  submateq  33318  cplgredgex  34638  acycgrcycl  34665