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

Theorem bdxor 12836
Description: The exclusive disjunction of two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
Hypotheses
Ref Expression
bdxor.1 BOUNDED 𝜑
bdxor.2 BOUNDED 𝜓
Assertion
Ref Expression
bdxor BOUNDED (𝜑𝜓)

Proof of Theorem bdxor
StepHypRef Expression
1 bdxor.1 . . . 4 BOUNDED 𝜑
2 bdxor.2 . . . 4 BOUNDED 𝜓
31, 2ax-bdor 12816 . . 3 BOUNDED (𝜑𝜓)
41, 2ax-bdan 12815 . . . 4 BOUNDED (𝜑𝜓)
54ax-bdn 12817 . . 3 BOUNDED ¬ (𝜑𝜓)
63, 5ax-bdan 12815 . 2 BOUNDED ((𝜑𝜓) ∧ ¬ (𝜑𝜓))
7 df-xor 1337 . 2 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓)))
86, 7bd0r 12825 1 BOUNDED (𝜑𝜓)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103  wo 680  wxo 1336  BOUNDED wbd 12812
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 12813  ax-bdan 12815  ax-bdor 12816  ax-bdn 12817
This theorem depends on definitions:  df-bi 116  df-xor 1337
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator