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Theorem vtocl2ga 3567
Description: Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 20-Aug-1995.) Avoid ax-10 2136 and ax-11 2153. (Revised by Gino Giotto, 20-Aug-2023.) (Proof shortened by Wolf Lammen, 23-Aug-2023.)
Hypotheses
Ref Expression
vtocl2ga.1 (𝑥 = 𝐴 → (𝜑𝜓))
vtocl2ga.2 (𝑦 = 𝐵 → (𝜓𝜒))
vtocl2ga.3 ((𝑥𝐶𝑦𝐷) → 𝜑)
Assertion
Ref Expression
vtocl2ga ((𝐴𝐶𝐵𝐷) → 𝜒)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦   𝜓,𝑥   𝜒,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)   𝜒(𝑥)   𝐵(𝑥)

Proof of Theorem vtocl2ga
StepHypRef Expression
1 vtocl2ga.2 . . . 4 (𝑦 = 𝐵 → (𝜓𝜒))
21imbi2d 339 . . 3 (𝑦 = 𝐵 → ((𝐴𝐶𝜓) ↔ (𝐴𝐶𝜒)))
3 vtocl2ga.1 . . . . . 6 (𝑥 = 𝐴 → (𝜑𝜓))
43imbi2d 339 . . . . 5 (𝑥 = 𝐴 → ((𝑦𝐷𝜑) ↔ (𝑦𝐷𝜓)))
5 vtocl2ga.3 . . . . . 6 ((𝑥𝐶𝑦𝐷) → 𝜑)
65ex 412 . . . . 5 (𝑥𝐶 → (𝑦𝐷𝜑))
74, 6vtoclga 3566 . . . 4 (𝐴𝐶 → (𝑦𝐷𝜓))
87com12 32 . . 3 (𝑦𝐷 → (𝐴𝐶𝜓))
92, 8vtoclga 3566 . 2 (𝐵𝐷 → (𝐴𝐶𝜒))
109impcom 407 1 ((𝐴𝐶𝐵𝐷) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1540  wcel 2105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809
This theorem is referenced by:  solin  5614  caovcan  7614  xpord2pred  8134  pwfseqlem2  10657  mulcanenq  10958  ltaddnq  10972  ltrnq  10977  genpv  10997  wrdind  14677  fsumrelem  15758  imasleval  17492  fullfunc  17862  fthfunc  17863  symggrplem  18802  pf1ind  22095  mretopd  22817  dvlip  25743  scvxcvx  26723  issubgoilem  30777  cnre2csqlem  33185
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