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Theorem vtocl2g 2816
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 2332 . 2 𝑥𝐴
2 nfcv 2332 . 2 𝑦𝐴
3 nfcv 2332 . 2 𝑦𝐵
4 nfv 1539 . 2 𝑥𝜓
5 nfv 1539 . 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 2814 1 ((𝐴𝑉𝐵𝑊) → 𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wcel 2160
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754
This theorem is referenced by:  uniprg  3839  intprg  3892  opthg  4256  opelopabsb  4278  unexb  4460  vtoclr  4692  elimasng  5014  cnvsng  5132  funopg  5269  f1osng  5521  fsng  5710  fvsng  5733  op1stg  6176  op2ndg  6177  xpsneng  6849  xpcomeng  6855  mhmlem  13071  bdunexb  15150  bj-unexg  15151
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