ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  vtocl2g GIF version

Theorem vtocl2g 2750
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 2281 . 2 𝑥𝐴
2 nfcv 2281 . 2 𝑦𝐴
3 nfcv 2281 . 2 𝑦𝐵
4 nfv 1508 . 2 𝑥𝜓
5 nfv 1508 . 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 2748 1 ((𝐴𝑉𝐵𝑊) → 𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wcel 1480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688
This theorem is referenced by:  uniprg  3751  intprg  3804  opthg  4160  opelopabsb  4182  unexb  4363  vtoclr  4587  elimasng  4907  cnvsng  5024  funopg  5157  f1osng  5408  fsng  5593  fvsng  5616  op1stg  6048  op2ndg  6049  xpsneng  6716  xpcomeng  6722  bdunexb  13225  bj-unexg  13226
  Copyright terms: Public domain W3C validator