Theorem orandc 929

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		pm4.56 770	. 2 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑 ∨ 𝜓))
2		dcn 832	. . . . 5 ⊢ (DECID 𝜑 → DECID ¬ 𝜑)
3	2	adantr 274	. . . 4 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → DECID ¬ 𝜑)
4		dcn 832	. . . . 5 ⊢ (DECID 𝜓 → DECID ¬ 𝜓)
5	4	adantl 275	. . . 4 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → DECID ¬ 𝜓)
6		dcan2 924	. . . 4 ⊢ (DECID ¬ 𝜑 → (DECID ¬ 𝜓 → DECID (¬ 𝜑 ∧ ¬ 𝜓)))
7	3, 5, 6	sylc 62	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → DECID (¬ 𝜑 ∧ ¬ 𝜓))
8		dcor 925	. . . 4 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 ∨ 𝜓)))
9	8	imp 123	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → DECID (𝜑 ∨ 𝜓))
10		con2bidc 865	. . 3 ⊢ (DECID (¬ 𝜑 ∧ ¬ 𝜓) → (DECID (𝜑 ∨ 𝜓) → (((¬ 𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑 ∨ 𝜓)) ↔ ((𝜑 ∨ 𝜓) ↔ ¬ (¬ 𝜑 ∧ ¬ 𝜓)))))
11	7, 9, 10	sylc 62	. 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (((¬ 𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑 ∨ 𝜓)) ↔ ((𝜑 ∨ 𝜓) ↔ ¬ (¬ 𝜑 ∧ ¬ 𝜓))))
12	1, 11	mpbii 147	1 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((𝜑 ∨ 𝜓) ↔ ¬ (¬ 𝜑 ∧ ¬ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698  DECID wdc 824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825

This theorem is referenced by:  gcdaddm  11917