Theorem con1bdc 878

Description: Contraposition. Bidirectional version of con1dc 856. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
con1bdc	⊢ (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑 → 𝜓) ↔ (¬ 𝜓 → 𝜑))))

Proof of Theorem con1bdc

Step	Hyp	Ref	Expression
1		con1dc 856	. . . 4 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)))
2	1	adantr 276	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)))
3		con1dc 856	. . . 4 ⊢ (DECID 𝜓 → ((¬ 𝜓 → 𝜑) → (¬ 𝜑 → 𝜓)))
4	3	adantl 277	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((¬ 𝜓 → 𝜑) → (¬ 𝜑 → 𝜓)))
5	2, 4	impbid 129	. 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((¬ 𝜑 → 𝜓) ↔ (¬ 𝜓 → 𝜑)))
6	5	ex 115	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑 → 𝜓) ↔ (¬ 𝜓 → 𝜑))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 834

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835

This theorem is referenced by: (None)