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Mirrors > Home > MPE Home > Th. List > 3orim123d | Structured version Visualization version GIF version |
Description: Deduction joining 3 implications to form implication of disjunctions. (Contributed by NM, 4-Apr-1997.) |
Ref | Expression |
---|---|
3anim123d.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
3anim123d.2 | ⊢ (𝜑 → (𝜃 → 𝜏)) |
3anim123d.3 | ⊢ (𝜑 → (𝜂 → 𝜁)) |
Ref | Expression |
---|---|
3orim123d | ⊢ (𝜑 → ((𝜓 ∨ 𝜃 ∨ 𝜂) → (𝜒 ∨ 𝜏 ∨ 𝜁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3anim123d.1 | . . . 4 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
2 | 3anim123d.2 | . . . 4 ⊢ (𝜑 → (𝜃 → 𝜏)) | |
3 | 1, 2 | orim12d 961 | . . 3 ⊢ (𝜑 → ((𝜓 ∨ 𝜃) → (𝜒 ∨ 𝜏))) |
4 | 3anim123d.3 | . . 3 ⊢ (𝜑 → (𝜂 → 𝜁)) | |
5 | 3, 4 | orim12d 961 | . 2 ⊢ (𝜑 → (((𝜓 ∨ 𝜃) ∨ 𝜂) → ((𝜒 ∨ 𝜏) ∨ 𝜁))) |
6 | df-3or 1085 | . 2 ⊢ ((𝜓 ∨ 𝜃 ∨ 𝜂) ↔ ((𝜓 ∨ 𝜃) ∨ 𝜂)) | |
7 | df-3or 1085 | . 2 ⊢ ((𝜒 ∨ 𝜏 ∨ 𝜁) ↔ ((𝜒 ∨ 𝜏) ∨ 𝜁)) | |
8 | 5, 6, 7 | 3imtr4g 296 | 1 ⊢ (𝜑 → ((𝜓 ∨ 𝜃 ∨ 𝜂) → (𝜒 ∨ 𝜏 ∨ 𝜁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 844 ∨ w3o 1083 |
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 206 df-an 396 df-or 845 df-3or 1085 |
This theorem is referenced by: fr3nr 7756 soxp 8115 poxp3 8136 zorn2lem6 10498 fpwwe2lem11 10638 fpwwe2lem12 10639 sltres 27550 colinearalglem4 28675 colinearxfr 35580 fin2so 36988 frege133d 43092 el1fzopredsuc 46605 fmtno4prmfac 46812 |
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