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Mirrors > Home > MPE Home > Th. List > 3orbi123i | Structured version Visualization version GIF version |
Description: Join 3 biconditionals with disjunction. (Contributed by NM, 17-May-1994.) |
Ref | Expression |
---|---|
bi3.1 | ⊢ (𝜑 ↔ 𝜓) |
bi3.2 | ⊢ (𝜒 ↔ 𝜃) |
bi3.3 | ⊢ (𝜏 ↔ 𝜂) |
Ref | Expression |
---|---|
3orbi123i | ⊢ ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ (𝜓 ∨ 𝜃 ∨ 𝜂)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bi3.1 | . . . 4 ⊢ (𝜑 ↔ 𝜓) | |
2 | bi3.2 | . . . 4 ⊢ (𝜒 ↔ 𝜃) | |
3 | 1, 2 | orbi12i 914 | . . 3 ⊢ ((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜃)) |
4 | bi3.3 | . . 3 ⊢ (𝜏 ↔ 𝜂) | |
5 | 3, 4 | orbi12i 914 | . 2 ⊢ (((𝜑 ∨ 𝜒) ∨ 𝜏) ↔ ((𝜓 ∨ 𝜃) ∨ 𝜂)) |
6 | df-3or 1089 | . 2 ⊢ ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ ((𝜑 ∨ 𝜒) ∨ 𝜏)) | |
7 | df-3or 1089 | . 2 ⊢ ((𝜓 ∨ 𝜃 ∨ 𝜂) ↔ ((𝜓 ∨ 𝜃) ∨ 𝜂)) | |
8 | 5, 6, 7 | 3bitr4i 303 | 1 ⊢ ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ (𝜓 ∨ 𝜃 ∨ 𝜂)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∨ wo 846 ∨ w3o 1087 |
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-or 847 df-3or 1089 |
This theorem is referenced by: ne3anior 3037 otthne 5487 brtp 5524 wecmpep 5669 cnvso 6288 sorpss 7718 epweon 7762 epweonALT 7763 soxp 8115 dford2 9615 elfz0lmr 13747 sltsolem1 27178 axlowdimlem6 28205 elxrge02 32098 dfon2 34764 frege129d 42514 dfxlim2 44564 |
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