Theorem pm3.14 742

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 884. The converse holds for decidable propositions, as seen at pm3.13dc 943. (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 621	. 2 ⊢ (¬ 𝜑 → ¬ (𝜑 ∧ 𝜓))
3		simpr 109	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜓)
4	3	con3i 621	. 2 ⊢ (¬ 𝜓 → ¬ (𝜑 ∧ 𝜓))
5	2, 4	jaoi 705	1 ⊢ ((¬ 𝜑 ∨ ¬ 𝜓) → ¬ (𝜑 ∧ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 697

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 603  ax-in2 604  ax-io 698

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  pm3.1  743  xoranor  1355  difindiss  3325