Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  vtoclb Structured version   Visualization version   GIF version

Theorem vtoclb 3254
 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 335 . 2 (𝑥 = 𝐴 → ((𝜑𝜓) ↔ (𝜒𝜃)))
5 vtoclb.4 . 2 (𝜑𝜓)
61, 4, 5vtocl 3250 1 (𝜒𝜃)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   = wceq 1480   ∈ wcel 1992  Vcvv 3191 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-12 2049  ax-ext 2606 This theorem depends on definitions:  df-bi 197  df-an 386  df-tru 1483  df-ex 1702  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-v 3193 This theorem is referenced by:  sbss  4061  bnj609  30687
 Copyright terms: Public domain W3C validator