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Theorem dfbi3dc 1285
Description: An alternate definition of the biconditional for decidable propositions. Theorem *5.23 of [WhiteheadRussell] p. 124, but with decidability conditions. (Contributed by Jim Kingdon, 5-May-2018.)
Assertion
Ref Expression
dfbi3dc (DECID φ → (DECID ψ → ((φψ) ↔ ((φ ψ) φ ¬ ψ)))))

Proof of Theorem dfbi3dc
StepHypRef Expression
1 dcn 745 . . . 4 (DECID ψDECID ¬ ψ)
2 xordc 1280 . . . . 5 (DECID φ → (DECID ¬ ψ → (¬ (φ ↔ ¬ ψ) ↔ ((φ ¬ ¬ ψ) ψ ¬ φ)))))
32imp 115 . . . 4 ((DECID φ DECID ¬ ψ) → (¬ (φ ↔ ¬ ψ) ↔ ((φ ¬ ¬ ψ) ψ ¬ φ))))
41, 3sylan2 270 . . 3 ((DECID φ DECID ψ) → (¬ (φ ↔ ¬ ψ) ↔ ((φ ¬ ¬ ψ) ψ ¬ φ))))
5 pm5.18dc 776 . . . 4 (DECID φ → (DECID ψ → ((φψ) ↔ ¬ (φ ↔ ¬ ψ))))
65imp 115 . . 3 ((DECID φ DECID ψ) → ((φψ) ↔ ¬ (φ ↔ ¬ ψ)))
7 notnotdc 765 . . . . . 6 (DECID ψ → (ψ ↔ ¬ ¬ ψ))
87anbi2d 437 . . . . 5 (DECID ψ → ((φ ψ) ↔ (φ ¬ ¬ ψ)))
9 ancom 253 . . . . . 6 ((¬ φ ¬ ψ) ↔ (¬ ψ ¬ φ))
109a1i 9 . . . . 5 (DECID ψ → ((¬ φ ¬ ψ) ↔ (¬ ψ ¬ φ)))
118, 10orbi12d 706 . . . 4 (DECID ψ → (((φ ψ) φ ¬ ψ)) ↔ ((φ ¬ ¬ ψ) ψ ¬ φ))))
1211adantl 262 . . 3 ((DECID φ DECID ψ) → (((φ ψ) φ ¬ ψ)) ↔ ((φ ¬ ¬ ψ) ψ ¬ φ))))
134, 6, 123bitr4d 209 . 2 ((DECID φ DECID ψ) → ((φψ) ↔ ((φ ψ) φ ¬ ψ))))
1413ex 108 1 (DECID φ → (DECID ψ → ((φψ) ↔ ((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  df-xor 1266
This theorem is referenced by:  pm5.24dc  1286
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