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Theorem vtocl2d 3508
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 (𝜑𝐴𝑉)
vtocl2d.b (𝜑𝐵𝑊)
vtocl2d.1 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜓𝜒))
vtocl2d.3 (𝜑𝜓)
Assertion
Ref Expression
vtocl2d (𝜑𝜒)
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
StepHypRef Expression
1 vtocl2d.b . 2 (𝜑𝐵𝑊)
2 vtocl2d.3 . . . 4 (𝜑𝜓)
32adantr 484 . . 3 ((𝜑𝑦 = 𝐵) → 𝜓)
4 vtocl2d.a . . . . 5 (𝜑𝐴𝑉)
5 vtocl2d.1 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜓𝜒))
65adantll 713 . . . . . 6 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝜓𝜒))
76pm5.74da 803 . . . . 5 ((𝜑𝑥 = 𝐴) → ((𝑦 = 𝐵𝜓) ↔ (𝑦 = 𝐵𝜒)))
82a1d 25 . . . . 5 (𝜑 → (𝑦 = 𝐵𝜓))
94, 7, 8vtocld 3507 . . . 4 (𝜑 → (𝑦 = 𝐵𝜒))
109imp 410 . . 3 ((𝜑𝑦 = 𝐵) → 𝜒)
113, 102thd 268 . 2 ((𝜑𝑦 = 𝐵) → (𝜓𝜒))
121, 11, 2vtocld 3507 1 (𝜑𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-nf 1786  df-cleq 2794  df-clel 2873  df-nfc 2941
This theorem is referenced by:  submateq  31162  cplgredgex  32475  acycgrcycl  32502
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