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Theorem vtocl2gaf 3579
Description: Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 10-Aug-2013.) (Proof shortened by Wolf Lammen, 31-May-2025.)
Hypotheses
Ref Expression
vtocl2gaf.a 𝑥𝐴
vtocl2gaf.b 𝑦𝐴
vtocl2gaf.c 𝑦𝐵
vtocl2gaf.1 𝑥𝜓
vtocl2gaf.2 𝑦𝜒
vtocl2gaf.3 (𝑥 = 𝐴 → (𝜑𝜓))
vtocl2gaf.4 (𝑦 = 𝐵 → (𝜓𝜒))
vtocl2gaf.5 ((𝑥𝐶𝑦𝐷) → 𝜑)
Assertion
Ref Expression
vtocl2gaf ((𝐴𝐶𝐵𝐷) → 𝜒)
Distinct variable groups:   𝑥,𝑦,𝐶   𝑥,𝐷,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem vtocl2gaf
StepHypRef Expression
1 vtocl2gaf.c . . 3 𝑦𝐵
2 vtocl2gaf.b . . . . 5 𝑦𝐴
32nfel1 2920 . . . 4 𝑦 𝐴𝐶
4 vtocl2gaf.2 . . . 4 𝑦𝜒
53, 4nfim 1894 . . 3 𝑦(𝐴𝐶𝜒)
6 vtocl2gaf.4 . . . 4 (𝑦 = 𝐵 → (𝜓𝜒))
76imbi2d 340 . . 3 (𝑦 = 𝐵 → ((𝐴𝐶𝜓) ↔ (𝐴𝐶𝜒)))
8 vtocl2gaf.a . . . . 5 𝑥𝐴
9 nfv 1912 . . . . . 6 𝑥 𝑦𝐷
10 vtocl2gaf.1 . . . . . 6 𝑥𝜓
119, 10nfim 1894 . . . . 5 𝑥(𝑦𝐷𝜓)
12 vtocl2gaf.3 . . . . . 6 (𝑥 = 𝐴 → (𝜑𝜓))
1312imbi2d 340 . . . . 5 (𝑥 = 𝐴 → ((𝑦𝐷𝜑) ↔ (𝑦𝐷𝜓)))
14 vtocl2gaf.5 . . . . . 6 ((𝑥𝐶𝑦𝐷) → 𝜑)
1514ex 412 . . . . 5 (𝑥𝐶 → (𝑦𝐷𝜑))
168, 11, 13, 15vtoclgaf 3576 . . . 4 (𝐴𝐶 → (𝑦𝐷𝜓))
1716com12 32 . . 3 (𝑦𝐷 → (𝐴𝐶𝜓))
181, 5, 7, 17vtoclgaf 3576 . 2 (𝐵𝐷 → (𝐴𝐶𝜒))
1918impcom 407 1 ((𝐴𝐶𝐵𝐷) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wnf 1780  wcel 2106  wnfc 2888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-v 3480
This theorem is referenced by:  vtocl3gaf  3581  ovmpos  7581  ov2gf  7582  ov3  7596  pwfseqlem2  10697  cnmptcom  23702
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