Theorem con2biddc 774

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 128	. . . 4 ⊢ ((𝜓 ↔ ¬ 𝜒) ↔ (¬ 𝜒 ↔ 𝜓))
3	1, 2	syl6ib 150	. . 3 ⊢ (𝜑 → (DECID 𝜒 → (¬ 𝜒 ↔ 𝜓)))
4	3	con1biddc 770	. 2 ⊢ (𝜑 → (DECID 𝜒 → (¬ 𝜓 ↔ 𝜒)))
5		bicom 128	. 2 ⊢ ((¬ 𝜓 ↔ 𝜒) ↔ (𝜒 ↔ ¬ 𝜓))
6	4, 5	syl6ib 150	1 ⊢ (𝜑 → (DECID 𝜒 → (𝜒 ↔ ¬ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98  DECID wdc 742

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-dc 743

This theorem is referenced by:  anordc  863  xor3dc  1278