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Theorem pm5.17dc 809
Description: Two ways of stating exclusive-or which are equivalent for a decidable proposition. Based on theorem *5.17 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 16-Apr-2018.)
Assertion
Ref Expression
pm5.17dc (DECID ψ → (((φ ψ) ¬ (φ ψ)) ↔ (φ ↔ ¬ ψ)))

Proof of Theorem pm5.17dc
StepHypRef Expression
1 bicom 128 . 2 ((φ ↔ ¬ ψ) ↔ (¬ ψφ))
2 dfbi2 368 . . 3 ((¬ ψφ) ↔ ((¬ ψφ) (φ → ¬ ψ)))
3 orcom 646 . . . . 5 ((φ ψ) ↔ (ψ φ))
4 dfordc 790 . . . . 5 (DECID ψ → ((ψ φ) ↔ (¬ ψφ)))
53, 4syl5rbb 182 . . . 4 (DECID ψ → ((¬ ψφ) ↔ (φ ψ)))
6 imnan 623 . . . . 5 ((φ → ¬ ψ) ↔ ¬ (φ ψ))
76a1i 9 . . . 4 (DECID ψ → ((φ → ¬ ψ) ↔ ¬ (φ ψ)))
85, 7anbi12d 442 . . 3 (DECID ψ → (((¬ ψφ) (φ → ¬ ψ)) ↔ ((φ ψ) ¬ (φ ψ))))
92, 8syl5bb 181 . 2 (DECID ψ → ((¬ ψφ) ↔ ((φ ψ) ¬ (φ ψ))))
101, 9syl5rbb 182 1 (DECID ψ → (((φ ψ) ¬ (φ ψ)) ↔ (φ ↔ ¬ ψ)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 628  DECID wdc 741
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
This theorem is referenced by:  xor2dc  1278
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