Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 1159 | . 2 ⊢ ((𝜑 → 𝜓) ∧ (𝜒 → 𝜓) ∧ (𝜃 → 𝜓)) |
5 | 3jao 1279 | . 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜓) ∧ (𝜃 → 𝜓)) → ((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓)) | |
6 | 4, 5 | ax-mp 5 | 1 ⊢ ((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ w3o 961 ∧ w3a 962 |
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-io 698 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 |
This theorem is referenced by: 3jaoian 1283 3ianorr 1287 acexmidlem1 5770 nndceq 6395 nndcel 6396 znegcl 9085 xrltnr 9566 nltpnft 9597 ngtmnft 9600 xrrebnd 9602 xnegcl 9615 xnegneg 9616 xltnegi 9618 xrpnfdc 9625 xrmnfdc 9626 xnegid 9642 xaddid1 9645 xposdif 9665 |
Copyright terms: Public domain | W3C validator |