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Theorem pm3.14 670
 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 799. The converse holds for decidable propositions, as seen at pm3.13dc 866. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
Assertion
Ref Expression
pm3.14 ((¬ 𝜑 ∨ ¬ 𝜓) → ¬ (𝜑𝜓))

Proof of Theorem pm3.14
StepHypRef Expression
1 simpl 102 . . 3 ((𝜑𝜓) → 𝜑)
21con3i 562 . 2 𝜑 → ¬ (𝜑𝜓))
3 simpr 103 . . 3 ((𝜑𝜓) → 𝜓)
43con3i 562 . 2 𝜓 → ¬ (𝜑𝜓))
52, 4jaoi 636 1 ((¬ 𝜑 ∨ ¬ 𝜓) → ¬ (𝜑𝜓))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ∨ wo 629 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 This theorem is referenced by:  pm3.1  671  xoranor  1268  difindiss  3191
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