Theorem pm5.15dc 1384

Description: A decidable proposition is equivalent to a decidable proposition or its negation. Based on theorem *5.15 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 18-Apr-2018.)

Assertion

Ref	Expression
pm5.15dc	⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ↔ 𝜓) ∨ (𝜑 ↔ ¬ 𝜓))))

Proof of Theorem pm5.15dc

Step	Hyp	Ref	Expression
1		xor3dc 1382	. . . . 5 ⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓))))
2	1	imp 123	. . . 4 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓)))
3	2	biimpd 143	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (¬ (𝜑 ↔ 𝜓) → (𝜑 ↔ ¬ 𝜓)))
4		dcbi 931	. . . . 5 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 ↔ 𝜓)))
5	4	imp 123	. . . 4 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → DECID (𝜑 ↔ 𝜓))
6		dfordc 887	. . . 4 ⊢ (DECID (𝜑 ↔ 𝜓) → (((𝜑 ↔ 𝜓) ∨ (𝜑 ↔ ¬ 𝜓)) ↔ (¬ (𝜑 ↔ 𝜓) → (𝜑 ↔ ¬ 𝜓))))
7	5, 6	syl 14	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (((𝜑 ↔ 𝜓) ∨ (𝜑 ↔ ¬ 𝜓)) ↔ (¬ (𝜑 ↔ 𝜓) → (𝜑 ↔ ¬ 𝜓))))
8	3, 7	mpbird 166	. 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((𝜑 ↔ 𝜓) ∨ (𝜑 ↔ ¬ 𝜓)))
9	8	ex 114	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ↔ 𝜓) ∨ (𝜑 ↔ ¬ 𝜓))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 703  DECID wdc 829

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-in1 609  ax-in2 610  ax-io 704

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830

This theorem is referenced by: (None)