Theorem con2bidc 876

Description: Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.)

Assertion

Ref	Expression
con2bidc	⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑))))

Proof of Theorem con2bidc

Step	Hyp	Ref	Expression
1		con1bidc 875	. . . . 5 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑 ↔ 𝜓) ↔ (¬ 𝜓 ↔ 𝜑))))
2	1	imp 124	. . . 4 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((¬ 𝜑 ↔ 𝜓) ↔ (¬ 𝜓 ↔ 𝜑)))
3		bicom 140	. . . 4 ⊢ ((¬ 𝜑 ↔ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑))
4		bicom 140	. . . 4 ⊢ ((¬ 𝜓 ↔ 𝜑) ↔ (𝜑 ↔ ¬ 𝜓))
5	2, 3, 4	3bitr3g 222	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((𝜓 ↔ ¬ 𝜑) ↔ (𝜑 ↔ ¬ 𝜓)))
6	5	bicomd 141	. 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((𝜑 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑)))
7	6	ex 115	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 835

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 615  ax-in2 616  ax-io 710

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836

This theorem is referenced by:  annimdc  939  pm4.55dc  940  orandc  941  nbbndc  1405