Theorem con1bidc 768

Description: Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.)

Assertion

Ref	Expression
con1bidc	⊢ (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑 ↔ 𝜓) ↔ (¬ 𝜓 ↔ 𝜑))))

Proof of Theorem con1bidc

Step	Hyp	Ref	Expression
1		con1biimdc 767	. . . 4 ⊢ (DECID 𝜑 → ((¬ 𝜑 ↔ 𝜓) → (¬ 𝜓 ↔ 𝜑)))
2	1	adantr 261	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((¬ 𝜑 ↔ 𝜓) → (¬ 𝜓 ↔ 𝜑)))
3		con1biimdc 767	. . . 4 ⊢ (DECID 𝜓 → ((¬ 𝜓 ↔ 𝜑) → (¬ 𝜑 ↔ 𝜓)))
4	3	adantl 262	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((¬ 𝜓 ↔ 𝜑) → (¬ 𝜑 ↔ 𝜓)))
5	2, 4	impbid 120	. 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((¬ 𝜑 ↔ 𝜓) ↔ (¬ 𝜓 ↔ 𝜑)))
6	5	ex 108	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑 ↔ 𝜓) ↔ (¬ 𝜓 ↔ 𝜑))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98  DECID wdc 742

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 630

This theorem depends on definitions:  df-bi 110  df-dc 743

This theorem is referenced by:  con2bidc  769