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

Theorem vtoclb 2676
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 23-Dec-1993.)
Hypotheses
Ref Expression
vtoclb.1 𝐴 ∈ V
vtoclb.2 (𝑥 = 𝐴 → (𝜑𝜒))
vtoclb.3 (𝑥 = 𝐴 → (𝜓𝜃))
vtoclb.4 (𝜑𝜓)
Assertion
Ref Expression
vtoclb (𝜒𝜃)
Distinct variable groups:   𝑥,𝐴   𝜒,𝑥   𝜃,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem vtoclb
StepHypRef Expression
1 vtoclb.1 . 2 𝐴 ∈ V
2 vtoclb.2 . . 3 (𝑥 = 𝐴 → (𝜑𝜒))
3 vtoclb.3 . . 3 (𝑥 = 𝐴 → (𝜓𝜃))
42, 3bibi12d 233 . 2 (𝑥 = 𝐴 → ((𝜑𝜓) ↔ (𝜒𝜃)))
5 vtoclb.4 . 2 (𝜑𝜓)
61, 4, 5vtocl 2673 1 (𝜒𝜃)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1289  wcel 1438  Vcvv 2619
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-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621
This theorem is referenced by:  alexeq  2741  sbss  3386
  Copyright terms: Public domain W3C validator