Theorem con1bdc 805

Description: Contraposition. Bidirectional version of con1dc 786. (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 786	. . . 4 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)))
2	1	adantr 270	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)))
3		con1dc 786	. . . 4 ⊢ (DECID 𝜓 → ((¬ 𝜓 → 𝜑) → (¬ 𝜑 → 𝜓)))
4	3	adantl 271	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((¬ 𝜓 → 𝜑) → (¬ 𝜑 → 𝜓)))
5	2, 4	impbid 127	. 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((¬ 𝜑 → 𝜓) ↔ (¬ 𝜓 → 𝜑)))
6	5	ex 113	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑 → 𝜓) ↔ (¬ 𝜓 → 𝜑))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103  DECID wdc 775

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115  df-dc 776

This theorem is referenced by: (None)