Theorem orandc 941

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		pm4.56 781	. 2 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑 ∨ 𝜓))
2		dcn 843	. . . 4 ⊢ (DECID 𝜑 → DECID ¬ 𝜑)
3		dcn 843	. . . 4 ⊢ (DECID 𝜓 → DECID ¬ 𝜓)
4		dcan 935	. . . 4 ⊢ ((DECID ¬ 𝜑 ∧ DECID ¬ 𝜓) → DECID (¬ 𝜑 ∧ ¬ 𝜓))
5	2, 3, 4	syl2an 289	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → DECID (¬ 𝜑 ∧ ¬ 𝜓))
6		dcor 937	. . . 4 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 ∨ 𝜓)))
7	6	imp 124	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → DECID (𝜑 ∨ 𝜓))
8		con2bidc 876	. . 3 ⊢ (DECID (¬ 𝜑 ∧ ¬ 𝜓) → (DECID (𝜑 ∨ 𝜓) → (((¬ 𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑 ∨ 𝜓)) ↔ ((𝜑 ∨ 𝜓) ↔ ¬ (¬ 𝜑 ∧ ¬ 𝜓)))))
9	5, 7, 8	sylc 62	. 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (((¬ 𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑 ∨ 𝜓)) ↔ ((𝜑 ∨ 𝜓) ↔ ¬ (¬ 𝜑 ∧ ¬ 𝜓))))
10	1, 9	mpbii 148	1 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((𝜑 ∨ 𝜓) ↔ ¬ (¬ 𝜑 ∧ ¬ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709  DECID wdc 835

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-in1 615  ax-in2 616  ax-io 710

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836

This theorem is referenced by:  gcdaddm  12151