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Theorem vtocl2g 2801
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 2319 . 2 𝑥𝐴
2 nfcv 2319 . 2 𝑦𝐴
3 nfcv 2319 . 2 𝑦𝐵
4 nfv 1528 . 2 𝑥𝜓
5 nfv 1528 . 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 2799 1 ((𝐴𝑉𝐵𝑊) → 𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739
This theorem is referenced by:  uniprg  3824  intprg  3877  opthg  4238  opelopabsb  4260  unexb  4442  vtoclr  4674  elimasng  4996  cnvsng  5114  funopg  5250  f1osng  5502  fsng  5689  fvsng  5712  op1stg  6150  op2ndg  6151  xpsneng  6821  xpcomeng  6827  mhmlem  12977  bdunexb  14642  bj-unexg  14643
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