Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bdcdeq GIF version

Theorem bdcdeq 13121
Description: Conditional equality of a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
Hypothesis
Ref Expression
bdcdeq.1 BOUNDED 𝜑
Assertion
Ref Expression
bdcdeq BOUNDED CondEq(𝑥 = 𝑦𝜑)

Proof of Theorem bdcdeq
StepHypRef Expression
1 ax-bdeq 13102 . . 3 BOUNDED 𝑥 = 𝑦
2 bdcdeq.1 . . 3 BOUNDED 𝜑
31, 2ax-bdim 13096 . 2 BOUNDED (𝑥 = 𝑦𝜑)
4 df-cdeq 2893 . 2 (CondEq(𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑦𝜑))
53, 4bd0r 13107 1 BOUNDED CondEq(𝑥 = 𝑦𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  CondEqwcdeq 2892  BOUNDED wbd 13094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-bd0 13095  ax-bdim 13096  ax-bdeq 13102
This theorem depends on definitions:  df-bi 116  df-cdeq 2893
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator