Theorem 3ianorr 1304

Description: Triple disjunction implies negated triple conjunction. (Contributed by Jim Kingdon, 23-Dec-2018.)

Assertion

Ref	Expression
3ianorr	⊢ ((¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒) → ¬ (𝜑 ∧ 𝜓 ∧ 𝜒))

Proof of Theorem 3ianorr

Step	Hyp	Ref	Expression
1		simp1 992	. . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜑)
2	1	con3i 627	. 2 ⊢ (¬ 𝜑 → ¬ (𝜑 ∧ 𝜓 ∧ 𝜒))
3		simp2 993	. . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜓)
4	3	con3i 627	. 2 ⊢ (¬ 𝜓 → ¬ (𝜑 ∧ 𝜓 ∧ 𝜒))
5		simp3 994	. . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜒)
6	5	con3i 627	. 2 ⊢ (¬ 𝜒 → ¬ (𝜑 ∧ 𝜓 ∧ 𝜒))
7	2, 4, 6	3jaoi 1298	1 ⊢ ((¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒) → ¬ (𝜑 ∧ 𝜓 ∧ 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ∨ w3o 972   ∧ w3a 973

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 609  ax-in2 610  ax-io 704

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975

This theorem is referenced by:  funtpg  5249