Theorem pm3.14 753

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 899. The converse holds for decidable propositions, as seen at pm3.13dc 959. (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 109	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
2	1	con3i 632	. 2 ⊢ (¬ 𝜑 → ¬ (𝜑 ∧ 𝜓))
3		simpr 110	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜓)
4	3	con3i 632	. 2 ⊢ (¬ 𝜓 → ¬ (𝜑 ∧ 𝜓))
5	2, 4	jaoi 716	1 ⊢ ((¬ 𝜑 ∨ ¬ 𝜓) → ¬ (𝜑 ∧ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  pm3.1  754  xoranor  1377  difindiss  3389