Theorem pm3.13dc 865

Description: Theorem *3.13 of [WhiteheadRussell] p. 111, but for decidable propositions. The converse, pm3.14 669, holds for all propositions. (Contributed by Jim Kingdon, 22-Apr-2018.)

Assertion

Ref	Expression
pm3.13dc	⊢ (DECID φ → (DECID ψ → (¬ (φ ∧ ψ) → (¬ φ ∨ ¬ ψ))))

Proof of Theorem pm3.13dc

Step	Hyp	Ref	Expression
1		dcn 745	. . 3 ⊢ (DECID φ → DECID ¬ φ)
2		dcn 745	. . 3 ⊢ (DECID ψ → DECID ¬ ψ)
3		dcor 842	. . 3 ⊢ (DECID ¬ φ → (DECID ¬ ψ → DECID (¬ φ ∨ ¬ ψ)))
4	1, 2, 3	syl2im 34	. 2 ⊢ (DECID φ → (DECID ψ → DECID (¬ φ ∨ ¬ ψ)))
5		pm3.11dc 863	. 2 ⊢ (DECID φ → (DECID ψ → (¬ (¬ φ ∨ ¬ ψ) → (φ ∧ ψ))))
6		con1dc 752	. 2 ⊢ (DECID (¬ φ ∨ ¬ ψ) → ((¬ (¬ φ ∨ ¬ ψ) → (φ ∧ ψ)) → (¬ (φ ∧ ψ) → (¬ φ ∨ ¬ ψ))))
7	4, 5, 6	syl6c 60	1 ⊢ (DECID φ → (DECID ψ → (¬ (φ ∧ ψ) → (¬ φ ∨ ¬ ψ))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ∨ wo 628  DECID wdc 741

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 629

This theorem depends on definitions:  df-bi 110  df-dc 742

This theorem is referenced by: (None)