ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  vtocl2ga GIF version

Theorem vtocl2ga 2817
Description: Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 20-Aug-1995.)
Hypotheses
Ref Expression
vtocl2ga.1 (𝑥 = 𝐴 → (𝜑𝜓))
vtocl2ga.2 (𝑦 = 𝐵 → (𝜓𝜒))
vtocl2ga.3 ((𝑥𝐶𝑦𝐷) → 𝜑)
Assertion
Ref Expression
vtocl2ga ((𝐴𝐶𝐵𝐷) → 𝜒)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦   𝜓,𝑥   𝜒,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)   𝜒(𝑥)   𝐵(𝑥)

Proof of Theorem vtocl2ga
StepHypRef Expression
1 nfcv 2329 . 2 𝑥𝐴
2 nfcv 2329 . 2 𝑦𝐴
3 nfcv 2329 . 2 𝑦𝐵
4 nfv 1538 . 2 𝑥𝜓
5 nfv 1538 . 2 𝑦𝜒
6 vtocl2ga.1 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
7 vtocl2ga.2 . 2 (𝑦 = 𝐵 → (𝜓𝜒))
8 vtocl2ga.3 . 2 ((𝑥𝐶𝑦𝐷) → 𝜑)
91, 2, 3, 4, 5, 6, 7, 8vtocl2gaf 2816 1 ((𝐴𝐶𝐵𝐷) → 𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1363  wcel 2158
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751
This theorem is referenced by:  caovcan  6052  genipv  7521  fsumrelem  11492
  Copyright terms: Public domain W3C validator