Theorem con1biddc 888

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 885	. 2 ⊢ (DECID 𝜓 → ((¬ 𝜓 ↔ 𝜒) → (¬ 𝜒 ↔ 𝜓)))
3	1, 2	sylcom 28	1 ⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜒 ↔ 𝜓)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 105  DECID wdc 846

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847

This theorem is used by:  con2biddc  892  pm5.18dc  895  necon1abiddc  2482  necon1bbiddc  2483  eupth2lem3lem4fi  16726