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Theorem vtocl2g 2684
Description: Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 25-Apr-1995.)
Hypotheses
Ref Expression
vtocl2g.1 (𝑥 = 𝐴 → (𝜑𝜓))
vtocl2g.2 (𝑦 = 𝐵 → (𝜓𝜒))
vtocl2g.3 𝜑
Assertion
Ref Expression
vtocl2g ((𝐴𝑉𝐵𝑊) → 𝜒)
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴   𝑦,𝐵   𝜓,𝑥   𝜒,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)   𝜒(𝑥)   𝐵(𝑥)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem vtocl2g
StepHypRef Expression
1 nfcv 2229 . 2 𝑥𝐴
2 nfcv 2229 . 2 𝑦𝐴
3 nfcv 2229 . 2 𝑦𝐵
4 nfv 1467 . 2 𝑥𝜓
5 nfv 1467 . 2 𝑦𝜒
6 vtocl2g.1 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
7 vtocl2g.2 . 2 (𝑦 = 𝐵 → (𝜓𝜒))
8 vtocl2g.3 . 2 𝜑
91, 2, 3, 4, 5, 6, 7, 8vtocl2gf 2682 1 ((𝐴𝑉𝐵𝑊) → 𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1290  wcel 1439
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622
This theorem is referenced by:  uniprg  3674  intprg  3727  opthg  4074  opelopabsb  4096  unexb  4277  vtoclr  4499  elimasng  4813  cnvsng  4929  funopg  5061  f1osng  5307  fsng  5484  fvsng  5507  op1stg  5935  op2ndg  5936  xpsneng  6592  xpcomeng  6598  bdunexb  12084  bj-unexg  12085
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