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Theorem annimdc 844
Description: Express conjunction in terms of implication. The forward direction, annimim 781, is valid for all propositions, but as an equivalence, it requires a decidability condition. (Contributed by Jim Kingdon, 25-Apr-2018.)
Assertion
Ref Expression
annimdc (DECID φ → (DECID ψ → ((φ ¬ ψ) ↔ ¬ (φψ))))

Proof of Theorem annimdc
StepHypRef Expression
1 imandc 785 . . . 4 (DECID ψ → ((φψ) ↔ ¬ (φ ¬ ψ)))
21adantl 262 . . 3 ((DECID φ DECID ψ) → ((φψ) ↔ ¬ (φ ¬ ψ)))
3 dcim 783 . . . . 5 (DECID φ → (DECID ψDECID (φψ)))
43imp 115 . . . 4 ((DECID φ DECID ψ) → DECID (φψ))
5 dcn 745 . . . . . 6 (DECID ψDECID ¬ ψ)
6 dcan 841 . . . . . 6 (DECID φ → (DECID ¬ ψDECID (φ ¬ ψ)))
75, 6syl5 28 . . . . 5 (DECID φ → (DECID ψDECID (φ ¬ ψ)))
87imp 115 . . . 4 ((DECID φ DECID ψ) → DECID (φ ¬ ψ))
9 con2bidc 768 . . . 4 (DECID (φψ) → (DECID (φ ¬ ψ) → (((φψ) ↔ ¬ (φ ¬ ψ)) ↔ ((φ ¬ ψ) ↔ ¬ (φψ)))))
104, 8, 9sylc 56 . . 3 ((DECID φ DECID ψ) → (((φψ) ↔ ¬ (φ ¬ ψ)) ↔ ((φ ¬ ψ) ↔ ¬ (φψ))))
112, 10mpbid 135 . 2 ((DECID φ DECID ψ) → ((φ ¬ ψ) ↔ ¬ (φψ)))
1211ex 108 1 (DECID φ → (DECID ψ → ((φ ¬ ψ) ↔ ¬ (φψ))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98  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:  xordidc  1287
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