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

Theorem vtocl2g 2676
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 2225 . 2 𝑥𝐴
2 nfcv 2225 . 2 𝑦𝐴
3 nfcv 2225 . 2 𝑦𝐵
4 nfv 1464 . 2 𝑥𝜓
5 nfv 1464 . 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 2674 1 ((𝐴𝑉𝐵𝑊) → 𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1287  wcel 1436
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617
This theorem is referenced by:  uniprg  3651  intprg  3704  opthg  4038  opelopabsb  4060  unexb  4240  vtoclr  4453  elimasng  4764  cnvsng  4879  funopg  5010  f1osng  5251  fsng  5427  fvsng  5450  op1stg  5872  op2ndg  5873  xpsneng  6484  xpcomeng  6490  bdunexb  11241  bj-unexg  11242
  Copyright terms: Public domain W3C validator