Theorem con2biidc 773

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

Hypothesis

Ref	Expression
con2biidc.1	⊢ (DECID 𝜓 → (𝜑 ↔ ¬ 𝜓))

Assertion

Ref	Expression
con2biidc	⊢ (DECID 𝜓 → (𝜓 ↔ ¬ 𝜑))

Proof of Theorem con2biidc

Step	Hyp	Ref	Expression
1		con2biidc.1	. . . 4 ⊢ (DECID 𝜓 → (𝜑 ↔ ¬ 𝜓))
2	1	bicomd 129	. . 3 ⊢ (DECID 𝜓 → (¬ 𝜓 ↔ 𝜑))
3	2	con1biidc 771	. 2 ⊢ (DECID 𝜓 → (¬ 𝜑 ↔ 𝜓))
4	3	bicomd 129	1 ⊢ (DECID 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:  dfexdc  1390  nnedc  2211