Theorem pm5.15dc 1277

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 1275	. . . . 5 ⊢ (DECID φ → (DECID ψ → (¬ (φ ↔ ψ) ↔ (φ ↔ ¬ ψ))))
2	1	imp 115	. . . 4 ⊢ ((DECID φ ∧ DECID ψ) → (¬ (φ ↔ ψ) ↔ (φ ↔ ¬ ψ)))
3	2	biimpd 132	. . 3 ⊢ ((DECID φ ∧ DECID ψ) → (¬ (φ ↔ ψ) → (φ ↔ ¬ ψ)))
4		dcbi 843	. . . . 5 ⊢ (DECID φ → (DECID ψ → DECID (φ ↔ ψ)))
5	4	imp 115	. . . 4 ⊢ ((DECID φ ∧ DECID ψ) → DECID (φ ↔ ψ))
6		dfordc 790	. . . 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 156	. 2 ⊢ ((DECID φ ∧ DECID ψ) → ((φ ↔ ψ) ∨ (φ ↔ ¬ ψ)))
9	8	ex 108	1 ⊢ (DECID φ → (DECID ψ → ((φ ↔ ψ) ∨ (φ ↔ ¬ ψ))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 628  DECID wdc 741

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 629

This theorem depends on definitions:  df-bi 110  df-dc 742

This theorem is referenced by: (None)