Theorem con1biidc 804

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 799	. . 3 ⊢ (DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑))
2		con1biidc.1	. . . 4 ⊢ (DECID 𝜑 → (¬ 𝜑 ↔ 𝜓))
3	2	notbid 624	. . 3 ⊢ (DECID 𝜑 → (¬ ¬ 𝜑 ↔ ¬ 𝜓))
4	1, 3	bitrd 186	. 2 ⊢ (DECID 𝜑 → (𝜑 ↔ ¬ 𝜓))
5	4	bicomd 139	1 ⊢ (DECID 𝜑 → (¬ 𝜓 ↔ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 103  DECID wdc 775

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115  df-dc 776

This theorem is referenced by:  con2biidc  806  necon1abiidc  2305  necon1bbiidc  2306