Theorem con1biidc 815

Description: A contraposition inference. (Contributed by Jim Kingdon, 15-Mar-2018.)

Hypothesis

Ref	Expression
con1biidc.1	⊢ (DECID 𝜑 → (¬ 𝜑 ↔ 𝜓))

Assertion

Ref	Expression
con1biidc	⊢ (DECID 𝜑 → (¬ 𝜓 ↔ 𝜑))

Proof of Theorem con1biidc

Step	Hyp	Ref	Expression
1		notnotbdc 810	. . 3 ⊢ (DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑))
2		con1biidc.1	. . . 4 ⊢ (DECID 𝜑 → (¬ 𝜑 ↔ 𝜓))
3	2	notbid 633	. . 3 ⊢ (DECID 𝜑 → (¬ ¬ 𝜑 ↔ ¬ 𝜓))
4	1, 3	bitrd 187	. 2 ⊢ (DECID 𝜑 → (𝜑 ↔ ¬ 𝜓))
5	4	bicomd 140	1 ⊢ (DECID 𝜑 → (¬ 𝜓 ↔ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104  DECID wdc 786

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 584  ax-in2 585  ax-io 671

This theorem depends on definitions:  df-bi 116  df-dc 787

This theorem is referenced by:  con2biidc  817  necon1abiidc  2327  necon1bbiidc  2328