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Mirrors > Home > ILE Home > Th. List > 3ianorr | GIF version |
Description: Triple disjunction implies negated triple conjunction. (Contributed by Jim Kingdon, 23-Dec-2018.) |
Ref | Expression |
---|---|
3ianorr | ⊢ ((¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒) → ¬ (𝜑 ∧ 𝜓 ∧ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 949 | . . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜑) | |
2 | 1 | con3i 602 | . 2 ⊢ (¬ 𝜑 → ¬ (𝜑 ∧ 𝜓 ∧ 𝜒)) |
3 | simp2 950 | . . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜓) | |
4 | 3 | con3i 602 | . 2 ⊢ (¬ 𝜓 → ¬ (𝜑 ∧ 𝜓 ∧ 𝜒)) |
5 | simp3 951 | . . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜒) | |
6 | 5 | con3i 602 | . 2 ⊢ (¬ 𝜒 → ¬ (𝜑 ∧ 𝜓 ∧ 𝜒)) |
7 | 2, 4, 6 | 3jaoi 1249 | 1 ⊢ ((¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒) → ¬ (𝜑 ∧ 𝜓 ∧ 𝜒)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∨ w3o 929 ∧ w3a 930 |
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 584 ax-in2 585 ax-io 671 |
This theorem depends on definitions: df-bi 116 df-3or 931 df-3an 932 |
This theorem is referenced by: funtpg 5110 |
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