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Theorem pm5.54dc 826
Description: A conjunction is equivalent to one of its conjuncts, given a decidable conjunct. Based on theorem *5.54 of [WhiteheadRussell] p. 125. (Contributed by Jim Kingdon, 30-Mar-2018.)
Assertion
Ref Expression
pm5.54dc (DECID φ → (((φ ψ) ↔ φ) ((φ ψ) ↔ ψ)))

Proof of Theorem pm5.54dc
StepHypRef Expression
1 df-dc 742 . . 3 (DECID φ ↔ (φ ¬ φ))
2 simpr 103 . . . . 5 ((φ ψ) → ψ)
3 ax-ia3 101 . . . . 5 (φ → (ψ → (φ ψ)))
42, 3impbid2 131 . . . 4 (φ → ((φ ψ) ↔ ψ))
5 simpl 102 . . . . 5 ((φ ψ) → φ)
6 ax-in2 545 . . . . 5 φ → (φ → (φ ψ)))
75, 6impbid2 131 . . . 4 φ → ((φ ψ) ↔ φ))
84, 7orim12i 675 . . 3 ((φ ¬ φ) → (((φ ψ) ↔ ψ) ((φ ψ) ↔ φ)))
91, 8sylbi 114 . 2 (DECID φ → (((φ ψ) ↔ ψ) ((φ ψ) ↔ φ)))
109orcomd 647 1 (DECID φ → (((φ ψ) ↔ φ) ((φ ψ) ↔ 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-in2 545  ax-io 629
This theorem depends on definitions:  df-bi 110  df-dc 742
This theorem is referenced by: (None)
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