Theorem pm4.54dc 897

Description: Theorem *4.54 of [WhiteheadRussell] p. 120, for decidable propositions. One form of DeMorgan's law. (Contributed by Jim Kingdon, 2-May-2018.)

Assertion

Ref	Expression
pm4.54dc	⊢ (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑 ∧ 𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓))))

Proof of Theorem pm4.54dc

Step	Hyp	Ref	Expression
1		dcn 837	. . . . 5 ⊢ (DECID 𝜑 → DECID ¬ 𝜑)
2		dfandc 879	. . . . 5 ⊢ (DECID ¬ 𝜑 → (DECID 𝜓 → ((¬ 𝜑 ∧ 𝜓) ↔ ¬ (¬ 𝜑 → ¬ 𝜓))))
3	1, 2	syl 14	. . . 4 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑 ∧ 𝜓) ↔ ¬ (¬ 𝜑 → ¬ 𝜓))))
4	3	imp 123	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((¬ 𝜑 ∧ 𝜓) ↔ ¬ (¬ 𝜑 → ¬ 𝜓)))
5		pm4.66dc 896	. . . . 5 ⊢ (DECID 𝜑 → ((¬ 𝜑 → ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓)))
6	5	adantr 274	. . . 4 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((¬ 𝜑 → ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓)))
7	6	notbid 662	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (¬ (¬ 𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓)))
8	4, 7	bitrd 187	. 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((¬ 𝜑 ∧ 𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓)))
9	8	ex 114	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑 ∧ 𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 703  DECID wdc 829

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 609  ax-in2 610  ax-io 704

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830

This theorem is referenced by:  pm4.55dc  933