Theorem vtocl2d 3530

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 486	. . 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 727	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒))
7	6	pm5.74da 816	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝑦 = 𝐵 → 𝜓) ↔ (𝑦 = 𝐵 → 𝜒)))
8	2	a1d 26	. . . . 5 ⊢ (𝜑 → (𝑦 = 𝐵 → 𝜓))
9	4, 7, 8	vtocld 3529	. . . 4 ⊢ (𝜑 → (𝑦 = 𝐵 → 𝜒))
10	9	imp 412	. . 3 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → 𝜒)
11	3, 10	2thd 268	. 2 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒))
12	1, 11, 2	vtocld 3529	1 ⊢ (𝜑 → 𝜒)

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

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

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-clel 2840

This theorem is used by:  fpwwe2lem4  10630  pwfseqlem4  10658  submateq  34239  cplgredgex  35639  acycgrcycl  35652  disjimeceqim2  39487  eldisjim3  39497