Theorem con2biddc 773

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 769	. 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 741

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 629

This theorem depends on definitions:  df-bi 110  df-dc 742

This theorem is referenced by:  anordc  862  xor3dc  1275