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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 542 | . . 3 ⊢ ((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜃)) |
4 | bi3.3 | . . 3 ⊢ (𝜏 ↔ 𝜂) | |
5 | 3, 4 | orbi12i 542 | . 2 ⊢ (((𝜑 ∨ 𝜒) ∨ 𝜏) ↔ ((𝜓 ∨ 𝜃) ∨ 𝜂)) |
6 | df-3or 1032 | . 2 ⊢ ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ ((𝜑 ∨ 𝜒) ∨ 𝜏)) | |
7 | df-3or 1032 | . 2 ⊢ ((𝜓 ∨ 𝜃 ∨ 𝜂) ↔ ((𝜓 ∨ 𝜃) ∨ 𝜂)) | |
8 | 5, 6, 7 | 3bitr4i 291 | 1 ⊢ ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ (𝜓 ∨ 𝜃 ∨ 𝜂)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∨ wo 382 ∨ w3o 1030 |
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 196 df-or 384 df-3or 1032 |
This theorem is referenced by: ne3anior 2875 wecmpep 5030 cnvso 5591 sorpss 6840 ordon 6874 soxp 7177 dford2 8400 axlowdimlem6 25627 elxrge02 28971 brtp 30892 socnv 30908 dfon2 30941 sltsolem1 31067 frege129d 37074 elfz0lmr 40367 |
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