Theorem con1biimdc 806

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 117	. . 3 ⊢ ((¬ 𝜑 ↔ 𝜓) → (¬ 𝜑 → 𝜓))
2		con1dc 792	. . 3 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)))
3	1, 2	syl5 32	. 2 ⊢ (DECID 𝜑 → ((¬ 𝜑 ↔ 𝜓) → (¬ 𝜓 → 𝜑)))
4		bi2 129	. . . 4 ⊢ ((¬ 𝜑 ↔ 𝜓) → (𝜓 → ¬ 𝜑))
5	4	con2d 590	. . 3 ⊢ ((¬ 𝜑 ↔ 𝜓) → (𝜑 → ¬ 𝜓))
6	5	a1i 9	. 2 ⊢ (DECID 𝜑 → ((¬ 𝜑 ↔ 𝜓) → (𝜑 → ¬ 𝜓)))
7	3, 6	impbidd 126	1 ⊢ (DECID 𝜑 → ((¬ 𝜑 ↔ 𝜓) → (¬ 𝜓 ↔ 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104  DECID wdc 781

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666

This theorem depends on definitions:  df-bi 116  df-dc 782

This theorem is referenced by:  con1bidc  807  con1biddc  809