Theorem con2biddc 807

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 138	. . . 4 ⊢ ((𝜓 ↔ ¬ 𝜒) ↔ (¬ 𝜒 ↔ 𝜓))
3	1, 2	syl6ib 159	. . 3 ⊢ (𝜑 → (DECID 𝜒 → (¬ 𝜒 ↔ 𝜓)))
4	3	con1biddc 803	. 2 ⊢ (𝜑 → (DECID 𝜒 → (¬ 𝜓 ↔ 𝜒)))
5		bicom 138	. 2 ⊢ ((¬ 𝜓 ↔ 𝜒) ↔ (𝜒 ↔ ¬ 𝜓))
6	4, 5	syl6ib 159	1 ⊢ (𝜑 → (DECID 𝜒 → (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:  anordc  897  xor3dc  1318