Theorem orandc 880

Description: Disjunction in terms of conjunction (De Morgan's law), for decidable propositions. Compare Theorem *4.57 of [WhiteheadRussell] p. 120. (Contributed by Jim Kingdon, 13-Dec-2021.)

Assertion

Ref	Expression
orandc	⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((𝜑 ∨ 𝜓) ↔ ¬ (¬ 𝜑 ∧ ¬ 𝜓)))

Proof of Theorem orandc

Step	Hyp	Ref	Expression
1		pm4.56 839	. 2 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑 ∨ 𝜓))
2		dcn 779	. . . . 5 ⊢ (DECID 𝜑 → DECID ¬ 𝜑)
3	2	adantr 270	. . . 4 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → DECID ¬ 𝜑)
4		dcn 779	. . . . 5 ⊢ (DECID 𝜓 → DECID ¬ 𝜓)
5	4	adantl 271	. . . 4 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → DECID ¬ 𝜓)
6		dcan 875	. . . 4 ⊢ (DECID ¬ 𝜑 → (DECID ¬ 𝜓 → DECID (¬ 𝜑 ∧ ¬ 𝜓)))
7	3, 5, 6	sylc 61	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → DECID (¬ 𝜑 ∧ ¬ 𝜓))
8		dcor 876	. . . 4 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 ∨ 𝜓)))
9	8	imp 122	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → DECID (𝜑 ∨ 𝜓))
10		con2bidc 802	. . 3 ⊢ (DECID (¬ 𝜑 ∧ ¬ 𝜓) → (DECID (𝜑 ∨ 𝜓) → (((¬ 𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑 ∨ 𝜓)) ↔ ((𝜑 ∨ 𝜓) ↔ ¬ (¬ 𝜑 ∧ ¬ 𝜓)))))
11	7, 9, 10	sylc 61	. 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (((¬ 𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑 ∨ 𝜓)) ↔ ((𝜑 ∨ 𝜓) ↔ ¬ (¬ 𝜑 ∧ ¬ 𝜓))))
12	1, 11	mpbii 146	1 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((𝜑 ∨ 𝜓) ↔ ¬ (¬ 𝜑 ∧ ¬ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103   ∨ wo 661  DECID wdc 775

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115  df-dc 776

This theorem is referenced by:  gcdaddm  10375