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Theorem bdxor 11215
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 11195 . . 3 BOUNDED (𝜑𝜓)
41, 2ax-bdan 11194 . . . 4 BOUNDED (𝜑𝜓)
54ax-bdn 11196 . . 3 BOUNDED ¬ (𝜑𝜓)
63, 5ax-bdan 11194 . 2 BOUNDED ((𝜑𝜓) ∧ ¬ (𝜑𝜓))
7 df-xor 1310 . 2 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓)))
86, 7bd0r 11204 1 BOUNDED (𝜑𝜓)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 102  wo 662  wxo 1309  BOUNDED wbd 11191
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-bd0 11192  ax-bdan 11194  ax-bdor 11195  ax-bdn 11196
This theorem depends on definitions:  df-bi 115  df-xor 1310
This theorem is referenced by: (None)
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