Theorem con1biddc 871

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 868	. 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 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-stab 826  df-dc 830

This theorem is referenced by:  con2biddc  875  pm5.18dc  878  necon1abiddc  2402  necon1bbiddc  2403