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

Theorem bdxor 7063
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 7043 . . 3 BOUNDED (φ ψ)
41, 2ax-bdan 7042 . . . 4 BOUNDED (φ ψ)
54ax-bdn 7044 . . 3 BOUNDED ¬ (φ ψ)
63, 5ax-bdan 7042 . 2 BOUNDED ((φ ψ) ¬ (φ ψ))
7 df-xor 1252 . 2 ((φψ) ↔ ((φ ψ) ¬ (φ ψ)))
86, 7bd0r 7052 1 BOUNDED (φψ)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wa 97   wo 616  wxo 1251  BOUNDED wbd 7039
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-bd0 7040  ax-bdan 7042  ax-bdor 7043  ax-bdn 7044
This theorem depends on definitions:  df-bi 110  df-xor 1252
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator