Theorem condcOLD 839

Description: Obsolete proof of condc 838 as of 18-Nov-2023. (Contributed by Jim Kingdon, 13-Mar-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
condcOLD	⊢ (DECID 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑)))

Proof of Theorem condcOLD

Step	Hyp	Ref	Expression
1		df-dc 820	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		ax-1 6	. . . 4 ⊢ (𝜑 → (𝜓 → 𝜑))
3	2	a1d 22	. . 3 ⊢ (𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑)))
4		pm2.27 40	. . . 4 ⊢ (¬ 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → ¬ 𝜓))
5		ax-in2 604	. . . 4 ⊢ (¬ 𝜓 → (𝜓 → 𝜑))
6	4, 5	syl6 33	. . 3 ⊢ (¬ 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑)))
7	3, 6	jaoi 705	. 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑)))
8	1, 7	sylbi 120	1 ⊢ (DECID 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 697  DECID wdc 819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 604  ax-io 698

This theorem depends on definitions:  df-bi 116  df-dc 820

This theorem is referenced by: (None)