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Mirrors > Home > ILE Home > Th. List > 3jaoi | GIF version |
Description: Disjunction of 3 antecedents (inference). (Contributed by NM, 12-Sep-1995.) |
Ref | Expression |
---|---|
3jaoi.1 | ⊢ (𝜑 → 𝜓) |
3jaoi.2 | ⊢ (𝜒 → 𝜓) |
3jaoi.3 | ⊢ (𝜃 → 𝜓) |
Ref | Expression |
---|---|
3jaoi | ⊢ ((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3jaoi.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
2 | 3jaoi.2 | . . 3 ⊢ (𝜒 → 𝜓) | |
3 | 3jaoi.3 | . . 3 ⊢ (𝜃 → 𝜓) | |
4 | 1, 2, 3 | 3pm3.2i 1082 | . 2 ⊢ ((𝜑 → 𝜓) ∧ (𝜒 → 𝜓) ∧ (𝜃 → 𝜓)) |
5 | 3jao 1196 | . 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜓) ∧ (𝜃 → 𝜓)) → ((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓)) | |
6 | 4, 5 | ax-mp 7 | 1 ⊢ ((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ w3o 884 ∧ w3a 885 |
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-io 630 |
This theorem depends on definitions: df-bi 110 df-3or 886 df-3an 887 |
This theorem is referenced by: 3jaoian 1200 3ianorr 1204 sspsstrir 3046 acexmidlem1 5508 nndceq 6077 nndcel 6078 znegcl 8276 xrltnr 8701 nltpnft 8730 ngtmnft 8731 xrrebnd 8732 xnegcl 8745 xnegneg 8746 xltnegi 8748 |
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