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Mirrors > Home > NFE 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 1130 | . 2 ⊢ ((φ → ψ) ∧ (χ → ψ) ∧ (θ → ψ)) |
5 | 3jao 1243 | . 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 setvar class |
Syntax hints: → wi 4 ∨ w3o 933 ∧ w3a 934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 |
This theorem is referenced by: 3jaoian 1247 |
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