Theorem dfifp2dc 987

Description: Alternate definition of the conditional operator for decidable propositions. The value of if-(𝜑, 𝜓, 𝜒) is "if 𝜑 then 𝜓, and if not 𝜑 then 𝜒". This is the definition used in Section II.24 of [Church] p. 129 (Definition D12 page 132) (see comment of df-ifp 984). (Contributed by BJ, 22-Jun-2019.)

Assertion

Ref	Expression
dfifp2dc	⊢ (DECID 𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒))))

Proof of Theorem dfifp2dc

Step	Hyp	Ref	Expression
1		ifp2 986	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) → ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒)))
2		exmiddc 841	. . . . 5 ⊢ (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
3		simpl 109	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒))) → 𝜑)
4		simprl 529	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒))) → (𝜑 → 𝜓))
5	3, 4	jcai 311	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒))) → (𝜑 ∧ 𝜓))
6	5	orcd 738	. . . . . 6 ⊢ ((𝜑 ∧ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒))) → ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))
7		simpl 109	. . . . . . . 8 ⊢ ((¬ 𝜑 ∧ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒))) → ¬ 𝜑)
8		simprr 531	. . . . . . . 8 ⊢ ((¬ 𝜑 ∧ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒))) → (¬ 𝜑 → 𝜒))
9	7, 8	jcai 311	. . . . . . 7 ⊢ ((¬ 𝜑 ∧ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒))) → (¬ 𝜑 ∧ 𝜒))
10	9	olcd 739	. . . . . 6 ⊢ ((¬ 𝜑 ∧ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒))) → ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))
11	6, 10	jaoian 800	. . . . 5 ⊢ (((𝜑 ∨ ¬ 𝜑) ∧ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒))) → ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))
12	2, 11	sylan 283	. . . 4 ⊢ ((DECID 𝜑 ∧ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒))) → ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))
13		df-ifp 984	. . . 4 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))
14	12, 13	sylibr 134	. . 3 ⊢ ((DECID 𝜑 ∧ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒))) → if-(𝜑, 𝜓, 𝜒))
15	14	ex 115	. 2 ⊢ (DECID 𝜑 → (((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒)) → if-(𝜑, 𝜓, 𝜒)))
16	1, 15	impbid2 143	1 ⊢ (DECID 𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 713  DECID wdc 839  if-wif 983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 618  ax-io 714

This theorem depends on definitions:  df-bi 117  df-dc 840  df-ifp 984

This theorem is referenced by:  dfifp3dc  988  dfifp5dc  990  ifpdfbidc  991