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Theorem vtocl2ga 3566
Description: Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 20-Aug-1995.) Avoid ax-10 2137 and ax-11 2154. (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 340 . . 3 (𝑦 = 𝐵 → ((𝐴𝐶𝜓) ↔ (𝐴𝐶𝜒)))
3 vtocl2ga.1 . . . . . 6 (𝑥 = 𝐴 → (𝜑𝜓))
43imbi2d 340 . . . . 5 (𝑥 = 𝐴 → ((𝑦𝐷𝜑) ↔ (𝑦𝐷𝜓)))
5 vtocl2ga.3 . . . . . 6 ((𝑥𝐶𝑦𝐷) → 𝜑)
65ex 413 . . . . 5 (𝑥𝐶 → (𝑦𝐷𝜑))
74, 6vtoclga 3565 . . . 4 (𝐴𝐶 → (𝑦𝐷𝜓))
87com12 32 . . 3 (𝑦𝐷 → (𝐴𝐶𝜓))
92, 8vtoclga 3565 . 2 (𝐵𝐷 → (𝐴𝐶𝜒))
109impcom 408 1 ((𝐴𝐶𝐵𝐷) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810
This theorem is referenced by:  solin  5613  caovcan  7610  xpord2pred  8130  pwfseqlem2  10653  mulcanenq  10954  ltaddnq  10968  ltrnq  10973  genpv  10993  wrdind  14671  fsumrelem  15752  imasleval  17486  fullfunc  17856  fthfunc  17857  symggrplem  18764  pf1ind  21873  mretopd  22595  dvlip  25509  scvxcvx  26487  issubgoilem  30508  cnre2csqlem  32885
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