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Mirrors > Home > MPE Home > Th. List > an6 | Structured version Visualization version GIF version |
Description: Rearrangement of 6 conjuncts. (Contributed by NM, 13-Mar-1995.) |
Ref | Expression |
---|---|
an6 | ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏 ∧ 𝜂)) ↔ ((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜏) ∧ (𝜒 ∧ 𝜂))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | an4 861 | . . 3 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ ((𝜃 ∧ 𝜏) ∧ 𝜂)) ↔ (((𝜑 ∧ 𝜓) ∧ (𝜃 ∧ 𝜏)) ∧ (𝜒 ∧ 𝜂))) | |
2 | an4 861 | . . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜃 ∧ 𝜏)) ↔ ((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜏))) | |
3 | 2 | anbi1i 727 | . . 3 ⊢ ((((𝜑 ∧ 𝜓) ∧ (𝜃 ∧ 𝜏)) ∧ (𝜒 ∧ 𝜂)) ↔ (((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜏)) ∧ (𝜒 ∧ 𝜂))) |
4 | 1, 3 | bitri 263 | . 2 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ ((𝜃 ∧ 𝜏) ∧ 𝜂)) ↔ (((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜏)) ∧ (𝜒 ∧ 𝜂))) |
5 | df-3an 1033 | . . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒)) | |
6 | df-3an 1033 | . . 3 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜂) ↔ ((𝜃 ∧ 𝜏) ∧ 𝜂)) | |
7 | 5, 6 | anbi12i 729 | . 2 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏 ∧ 𝜂)) ↔ (((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ ((𝜃 ∧ 𝜏) ∧ 𝜂))) |
8 | df-3an 1033 | . 2 ⊢ (((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜏) ∧ (𝜒 ∧ 𝜂)) ↔ (((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜏)) ∧ (𝜒 ∧ 𝜂))) | |
9 | 4, 7, 8 | 3bitr4i 291 | 1 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏 ∧ 𝜂)) ↔ ((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜏) ∧ (𝜒 ∧ 𝜂))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∧ w3a 1031 |
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-an 385 df-3an 1033 |
This theorem is referenced by: 3an6 1401 elfzuzb 12207 fzadd2 12247 ptbasin 21190 iimulcl 22544 nb3grapr 25982 nb3grapr2 25983 txpcon 30468 paddasslem9 34132 paddasslem10 34133 gboage9 40186 nb3grpr 40610 nb3grpr2 40611 |
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