Theorem con1biimdc 767

Description: Contraposition. (Contributed by Jim Kingdon, 4-Apr-2018.)

Assertion

Ref	Expression
con1biimdc	⊢ (DECID 𝜑 → ((¬ 𝜑 ↔ 𝜓) → (¬ 𝜓 ↔ 𝜑)))

Proof of Theorem con1biimdc

Step	Hyp	Ref	Expression
1		bi1 111	. . 3 ⊢ ((¬ 𝜑 ↔ 𝜓) → (¬ 𝜑 → 𝜓))
2		con1dc 753	. . 3 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)))
3	1, 2	syl5 28	. 2 ⊢ (DECID 𝜑 → ((¬ 𝜑 ↔ 𝜓) → (¬ 𝜓 → 𝜑)))
4		bi2 121	. . . 4 ⊢ ((¬ 𝜑 ↔ 𝜓) → (𝜓 → ¬ 𝜑))
5	4	con2d 554	. . 3 ⊢ ((¬ 𝜑 ↔ 𝜓) → (𝜑 → ¬ 𝜓))
6	5	a1i 9	. 2 ⊢ (DECID 𝜑 → ((¬ 𝜑 ↔ 𝜓) → (𝜑 → ¬ 𝜓)))
7	3, 6	impbidd 118	1 ⊢ (DECID 𝜑 → ((¬ 𝜑 ↔ 𝜓) → (¬ 𝜓 ↔ 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ 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:  con1bidc  768  con1biddc  770