Theorem pm3.14 708

Description: Theorem *3.14 of [WhiteheadRussell] p. 111. One direction of De Morgan's law). The biconditional holds for decidable propositions as seen at ianordc 840. The converse holds for decidable propositions, as seen at pm3.13dc 908. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)

Assertion

Ref	Expression
pm3.14	⊢ ((¬ 𝜑 ∨ ¬ 𝜓) → ¬ (𝜑 ∧ 𝜓))

Proof of Theorem pm3.14

Step	Hyp	Ref	Expression
1		simpl 108	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
2	1	con3i 600	. 2 ⊢ (¬ 𝜑 → ¬ (𝜑 ∧ 𝜓))
3		simpr 109	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜓)
4	3	con3i 600	. 2 ⊢ (¬ 𝜓 → ¬ (𝜑 ∧ 𝜓))
5	2, 4	jaoi 674	1 ⊢ ((¬ 𝜑 ∨ ¬ 𝜓) → ¬ (𝜑 ∧ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 667

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  pm3.1  709  xoranor  1320  difindiss  3269