Theorem con1biddc 803

Description: A contraposition deduction. (Contributed by Jim Kingdon, 4-Apr-2018.)

Hypothesis

Ref	Expression
con1biddc.1	⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 ↔ 𝜒)))

Assertion

Ref	Expression
con1biddc	⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜒 ↔ 𝜓)))

Proof of Theorem con1biddc

Step	Hyp	Ref	Expression
1		con1biddc.1	. 2 ⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 ↔ 𝜒)))
2		con1biimdc 800	. 2 ⊢ (DECID 𝜓 → ((¬ 𝜓 ↔ 𝜒) → (¬ 𝜒 ↔ 𝜓)))
3	1, 2	sylcom 28	1 ⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜒 ↔ 𝜓)))

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:  con2biddc  807  pm5.18dc  810  necon1abiddc  2307  necon1bbiddc  2308