Theorem pm5.15dc 1438

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 1436	. . . . 5 ⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓))))
2	1	imp 124	. . . 4 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓)))
3	2	biimpd 144	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (¬ (𝜑 ↔ 𝜓) → (𝜑 ↔ ¬ 𝜓)))
4		dcbi 949	. . . . 5 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 ↔ 𝜓)))
5	4	imp 124	. . . 4 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → DECID (𝜑 ↔ 𝜓))
6		dfordc 904	. . . 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 167	. 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((𝜑 ↔ 𝜓) ∨ (𝜑 ↔ ¬ 𝜓)))
9	8	ex 115	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ↔ 𝜓) ∨ (𝜑 ↔ ¬ 𝜓))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720  DECID wdc 846

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847

This theorem is used by: (None)