Theorem con2bidc 769

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 768	. . . . 5 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑 ↔ 𝜓) ↔ (¬ 𝜓 ↔ 𝜑))))
2	1	imp 115	. . . 4 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((¬ 𝜑 ↔ 𝜓) ↔ (¬ 𝜓 ↔ 𝜑)))
3		bicom 128	. . . 4 ⊢ ((¬ 𝜑 ↔ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑))
4		bicom 128	. . . 4 ⊢ ((¬ 𝜓 ↔ 𝜑) ↔ (𝜑 ↔ ¬ 𝜓))
5	2, 3, 4	3bitr3g 211	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((𝜓 ↔ ¬ 𝜑) ↔ (𝜑 ↔ ¬ 𝜓)))
6	5	bicomd 129	. 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((𝜑 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑)))
7	6	ex 108	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ 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:  annimdc  845  pm4.55dc  846  nbbndc  1285