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Theorem con1bdc 783
 Description: Contraposition. Bidirectional version of con1dc 764. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
con1bdc (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑𝜓) ↔ (¬ 𝜓𝜑))))

Proof of Theorem con1bdc
StepHypRef Expression
1 con1dc 764 . . . 4 (DECID 𝜑 → ((¬ 𝜑𝜓) → (¬ 𝜓𝜑)))
21adantr 265 . . 3 ((DECID 𝜑DECID 𝜓) → ((¬ 𝜑𝜓) → (¬ 𝜓𝜑)))
3 con1dc 764 . . . 4 (DECID 𝜓 → ((¬ 𝜓𝜑) → (¬ 𝜑𝜓)))
43adantl 266 . . 3 ((DECID 𝜑DECID 𝜓) → ((¬ 𝜓𝜑) → (¬ 𝜑𝜓)))
52, 4impbid 124 . 2 ((DECID 𝜑DECID 𝜓) → ((¬ 𝜑𝜓) ↔ (¬ 𝜓𝜑)))
65ex 112 1 (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑𝜓) ↔ (¬ 𝜓𝜑))))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 101   ↔ wb 102  DECID wdc 753 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640 This theorem depends on definitions:  df-bi 114  df-dc 754 This theorem is referenced by: (None)
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