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Theorem xordc 1280
Description: Two ways to express "exclusive or" between decidable propositions. Theorem *5.22 of [WhiteheadRussell] p. 124, but for decidable propositions. (Contributed by Jim Kingdon, 5-May-2018.)
Assertion
Ref Expression
xordc (DECID φ → (DECID ψ → (¬ (φψ) ↔ ((φ ¬ ψ) (ψ ¬ φ)))))

Proof of Theorem xordc
StepHypRef Expression
1 excxor 1268 . . . 4 ((φψ) ↔ ((φ ¬ ψ) φ ψ)))
2 ancom 253 . . . . 5 ((¬ φ ψ) ↔ (ψ ¬ φ))
32orbi2i 678 . . . 4 (((φ ¬ ψ) φ ψ)) ↔ ((φ ¬ ψ) (ψ ¬ φ)))
41, 3bitri 173 . . 3 ((φψ) ↔ ((φ ¬ ψ) (ψ ¬ φ)))
5 xornbidc 1279 . . . 4 (DECID φ → (DECID ψ → ((φψ) ↔ ¬ (φψ))))
65imp 115 . . 3 ((DECID φ DECID ψ) → ((φψ) ↔ ¬ (φψ)))
74, 6syl5rbbr 184 . 2 ((DECID φ DECID ψ) → (¬ (φψ) ↔ ((φ ¬ ψ) (ψ ¬ φ))))
87ex 108 1 (DECID φ → (DECID ψ → (¬ (φψ) ↔ ((φ ¬ ψ) (ψ ¬ φ)))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 628  DECID wdc 741  wxo 1265
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-in1 544  ax-in2 545  ax-io 629
This theorem depends on definitions:  df-bi 110  df-dc 742  df-xor 1266
This theorem is referenced by:  dfbi3dc  1285  pm5.24dc  1286
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