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| Mirrors > Home > MPE Home > Th. List > in12 | Structured version Visualization version GIF version | ||
| Description: A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.) |
| Ref | Expression |
|---|---|
| in12 | ⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐵 ∩ (𝐴 ∩ 𝐶)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | incom 4159 | . . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) | |
| 2 | 1 | ineq1i 4166 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐵 ∩ 𝐴) ∩ 𝐶) |
| 3 | inass 4177 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵 ∩ 𝐶)) | |
| 4 | inass 4177 | . 2 ⊢ ((𝐵 ∩ 𝐴) ∩ 𝐶) = (𝐵 ∩ (𝐴 ∩ 𝐶)) | |
| 5 | 2, 3, 4 | 3eqtr3i 2792 | 1 ⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐵 ∩ (𝐴 ∩ 𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1559 ∩ cin 3901 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-ext 2733 |
| This theorem depends on definitions: df-bi 209 df-an 400 df-tru 1562 df-ex 1799 df-sb 2090 df-clab 2740 df-cleq 2753 df-clel 2836 df-rab 3414 df-v 3455 df-in 3909 |
| This theorem is referenced by: in32 4179 in31 4181 in4 4183 resdmres 6214 kmlem12 10112 ressress 17274 fh1 31778 fh2 31779 mdslmd3i 32492 bj-inrab3 37375 |
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