Theorem pm4.79dc 809

Description: Equivalence between a disjunction of two implications, and a conjunction and an implication. Based on theorem *4.79 of [WhiteheadRussell] p. 121 but with additional decidability antecedents. (Contributed by Jim Kingdon, 28-Mar-2018.)

Assertion

Ref	Expression
pm4.79dc	⊢ (DECID 𝜑 → (DECID 𝜓 → (((𝜓 → 𝜑) ∨ (𝜒 → 𝜑)) ↔ ((𝜓 ∧ 𝜒) → 𝜑))))

Proof of Theorem pm4.79dc

Step	Hyp	Ref	Expression
1		id 19	. . . 4 ⊢ ((𝜓 → 𝜑) → (𝜓 → 𝜑))
2		id 19	. . . 4 ⊢ ((𝜒 → 𝜑) → (𝜒 → 𝜑))
3	1, 2	jaoa 640	. . 3 ⊢ (((𝜓 → 𝜑) ∨ (𝜒 → 𝜑)) → ((𝜓 ∧ 𝜒) → 𝜑))
4		simplimdc 757	. . . . . 6 ⊢ (DECID 𝜓 → (¬ (𝜓 → 𝜑) → 𝜓))
5		pm3.3 248	. . . . . 6 ⊢ (((𝜓 ∧ 𝜒) → 𝜑) → (𝜓 → (𝜒 → 𝜑)))
6	4, 5	syl9 66	. . . . 5 ⊢ (DECID 𝜓 → (((𝜓 ∧ 𝜒) → 𝜑) → (¬ (𝜓 → 𝜑) → (𝜒 → 𝜑))))
7		dcim 784	. . . . . 6 ⊢ (DECID 𝜓 → (DECID 𝜑 → DECID (𝜓 → 𝜑)))
8		pm2.54dc 790	. . . . . 6 ⊢ (DECID (𝜓 → 𝜑) → ((¬ (𝜓 → 𝜑) → (𝜒 → 𝜑)) → ((𝜓 → 𝜑) ∨ (𝜒 → 𝜑))))
9	7, 8	syl6 29	. . . . 5 ⊢ (DECID 𝜓 → (DECID 𝜑 → ((¬ (𝜓 → 𝜑) → (𝜒 → 𝜑)) → ((𝜓 → 𝜑) ∨ (𝜒 → 𝜑)))))
10	6, 9	syl5d 62	. . . 4 ⊢ (DECID 𝜓 → (DECID 𝜑 → (((𝜓 ∧ 𝜒) → 𝜑) → ((𝜓 → 𝜑) ∨ (𝜒 → 𝜑)))))
11	10	imp 115	. . 3 ⊢ ((DECID 𝜓 ∧ DECID 𝜑) → (((𝜓 ∧ 𝜒) → 𝜑) → ((𝜓 → 𝜑) ∨ (𝜒 → 𝜑))))
12	3, 11	impbid2 131	. 2 ⊢ ((DECID 𝜓 ∧ DECID 𝜑) → (((𝜓 → 𝜑) ∨ (𝜒 → 𝜑)) ↔ ((𝜓 ∧ 𝜒) → 𝜑)))
13	12	expcom 109	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → (((𝜓 → 𝜑) ∨ (𝜒 → 𝜑)) ↔ ((𝜓 ∧ 𝜒) → 𝜑))))

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-in1 544  ax-in2 545  ax-io 630

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

This theorem is referenced by: (None)