Theorem pm4.83dc 858

Description: Theorem *4.83 of [WhiteheadRussell] p. 122, for decidable propositions. As with other case elimination theorems, like pm2.61dc 762, it only holds for decidable propositions. (Contributed by Jim Kingdon, 12-May-2018.)

Assertion

Ref	Expression
pm4.83dc	⊢ (DECID 𝜑 → (((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜓)) ↔ 𝜓))

Proof of Theorem pm4.83dc

Step	Hyp	Ref	Expression
1		df-dc 743	. . 3 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		pm3.44 635	. . . 4 ⊢ (((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜓)) → ((𝜑 ∨ ¬ 𝜑) → 𝜓))
3	2	com12 27	. . 3 ⊢ ((𝜑 ∨ ¬ 𝜑) → (((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜓)) → 𝜓))
4	1, 3	sylbi 114	. 2 ⊢ (DECID 𝜑 → (((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜓)) → 𝜓))
5		ax-1 5	. . 3 ⊢ (𝜓 → (𝜑 → 𝜓))
6		ax-1 5	. . 3 ⊢ (𝜓 → (¬ 𝜑 → 𝜓))
7	5, 6	jca 290	. 2 ⊢ (𝜓 → ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜓)))
8	4, 7	impbid1 130	1 ⊢ (DECID 𝜑 → (((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜓)) ↔ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629  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-io 630

This theorem depends on definitions:  df-bi 110  df-dc 743

This theorem is referenced by: (None)