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

Theorem vtocl2ga 2621
 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 2178 . 2 𝑥𝐴
2 nfcv 2178 . 2 𝑦𝐴
3 nfcv 2178 . 2 𝑦𝐵
4 nfv 1421 . 2 𝑥𝜓
5 nfv 1421 . 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 2620 1 ((𝐴𝐶𝐵𝐷) → 𝜒)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559 This theorem is referenced by:  caovcan  5665  genipv  6607
 Copyright terms: Public domain W3C validator