Theorem con2biddc 880

Description: A contraposition deduction. (Contributed by Jim Kingdon, 11-Apr-2018.)

Hypothesis

Ref	Expression
con2biddc.1	⊢ (𝜑 → (DECID 𝜒 → (𝜓 ↔ ¬ 𝜒)))

Assertion

Ref	Expression
con2biddc	⊢ (𝜑 → (DECID 𝜒 → (𝜒 ↔ ¬ 𝜓)))

Proof of Theorem con2biddc

Step	Hyp	Ref	Expression
1		con2biddc.1	. . . 4 ⊢ (𝜑 → (DECID 𝜒 → (𝜓 ↔ ¬ 𝜒)))
2		bicom 140	. . . 4 ⊢ ((𝜓 ↔ ¬ 𝜒) ↔ (¬ 𝜒 ↔ 𝜓))
3	1, 2	syl6ib 161	. . 3 ⊢ (𝜑 → (DECID 𝜒 → (¬ 𝜒 ↔ 𝜓)))
4	3	con1biddc 876	. 2 ⊢ (𝜑 → (DECID 𝜒 → (¬ 𝜓 ↔ 𝜒)))
5		bicom 140	. 2 ⊢ ((¬ 𝜓 ↔ 𝜒) ↔ (𝜒 ↔ ¬ 𝜓))
6	4, 5	syl6ib 161	1 ⊢ (𝜑 → (DECID 𝜒 → (𝜒 ↔ ¬ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105  DECID wdc 834

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835

This theorem is referenced by:  anordc  956  xor3dc  1387