Theorem pm4.54dc 805

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 746	. . . . 5 ⊢ (DECID 𝜑 → DECID ¬ 𝜑)
2		dfandc 778	. . . . 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 115	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((¬ 𝜑 ∧ 𝜓) ↔ ¬ (¬ 𝜑 → ¬ 𝜓)))
5		pm4.66dc 802	. . . . 5 ⊢ (DECID 𝜑 → ((¬ 𝜑 → ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓)))
6	5	adantr 261	. . . 4 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((¬ 𝜑 → ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓)))
7	6	notbid 592	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (¬ (¬ 𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓)))
8	4, 7	bitrd 177	. 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((¬ 𝜑 ∧ 𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓)))
9	8	ex 108	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑 ∧ 𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629  DECID wdc 742

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-dc 743

This theorem is referenced by:  pm4.55dc  846