Users' Mathboxes Mathbox for Jeff Hoffman < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fveleq Structured version   Visualization version   GIF version

Theorem fveleq 33826
Description: Please add description here. (Contributed by Jeff Hoffman, 12-Feb-2008.)
Assertion
Ref Expression
fveleq (𝐴 = 𝐵 → ((𝜑 → (𝐹𝐴) ∈ 𝑃) ↔ (𝜑 → (𝐹𝐵) ∈ 𝑃)))

Proof of Theorem fveleq
StepHypRef Expression
1 fveq2 6659 . . 3 (𝐴 = 𝐵 → (𝐹𝐴) = (𝐹𝐵))
21eleq1d 2900 . 2 (𝐴 = 𝐵 → ((𝐹𝐴) ∈ 𝑃 ↔ (𝐹𝐵) ∈ 𝑃))
32imbi2d 344 1 (𝐴 = 𝐵 → ((𝜑 → (𝐹𝐴) ∈ 𝑃) ↔ (𝜑 → (𝐹𝐵) ∈ 𝑃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wcel 2115  cfv 6344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-iota 6303  df-fv 6352
This theorem is referenced by:  findfvcl  33827
  Copyright terms: Public domain W3C validator