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Theorem xordc1 1267
Description: Exclusive or implies the left proposition is decidable. (Contributed by Jim Kingdon, 12-Mar-2018.)
Assertion
Ref Expression
xordc1 ((φψ) → DECID φ)

Proof of Theorem xordc1
StepHypRef Expression
1 andir 720 . . 3 (((φ ψ) ¬ (φ ψ)) ↔ ((φ ¬ (φ ψ)) (ψ ¬ (φ ψ))))
2 ax-ia1 99 . . . 4 ((φ ¬ (φ ψ)) → φ)
3 imnan 611 . . . . . 6 ((ψ → ¬ φ) ↔ ¬ (ψ φ))
4 ancom 253 . . . . . 6 ((φ ψ) ↔ (ψ φ))
53, 4xchbinxr 595 . . . . 5 ((ψ → ¬ φ) ↔ ¬ (φ ψ))
6 pm3.35 329 . . . . 5 ((ψ (ψ → ¬ φ)) → ¬ φ)
75, 6sylan2br 272 . . . 4 ((ψ ¬ (φ ψ)) → ¬ φ)
82, 7orim12i 663 . . 3 (((φ ¬ (φ ψ)) (ψ ¬ (φ ψ))) → (φ ¬ φ))
91, 8sylbi 114 . 2 (((φ ψ) ¬ (φ ψ)) → (φ ¬ φ))
10 df-xor 1252 . 2 ((φψ) ↔ ((φ ψ) ¬ (φ ψ)))
11 df-dc 734 . 2 (DECID φ ↔ (φ ¬ φ))
129, 10, 113imtr4i 190 1 ((φψ) → DECID φ)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   wo 616  DECID wdc 733  wxo 1251
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 532  ax-in2 533  ax-io 617
This theorem depends on definitions:  df-bi 110  df-dc 734  df-xor 1252
This theorem is referenced by: (None)
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