Theorem con2biidc 874

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 140	. . 3 ⊢ (DECID 𝜓 → (¬ 𝜓 ↔ 𝜑))
3	2	con1biidc 872	. 2 ⊢ (DECID 𝜓 → (¬ 𝜑 ↔ 𝜓))
4	3	bicomd 140	1 ⊢ (DECID 𝜓 → (𝜓 ↔ ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104  DECID wdc 829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704

This theorem depends on definitions:  df-bi 116  df-dc 830

This theorem is referenced by:  dfexdc  1494  nnedc  2345