Theorem con1bdc 873

Description: Contraposition. Bidirectional version of con1dc 851. (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 851	. . . 4 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)))
2	1	adantr 274	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)))
3		con1dc 851	. . . 4 ⊢ (DECID 𝜓 → ((¬ 𝜓 → 𝜑) → (¬ 𝜑 → 𝜓)))
4	3	adantl 275	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((¬ 𝜓 → 𝜑) → (¬ 𝜑 → 𝜓)))
5	2, 4	impbid 128	. 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((¬ 𝜑 → 𝜓) ↔ (¬ 𝜓 → 𝜑)))
6	5	ex 114	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑 → 𝜓) ↔ (¬ 𝜓 → 𝜑))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104  DECID wdc 829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830

This theorem is referenced by: (None)