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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 4161 | . . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) | |
| 2 | 1 | ineq1i 4168 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐵 ∩ 𝐴) ∩ 𝐶) |
| 3 | inass 4179 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵 ∩ 𝐶)) | |
| 4 | inass 4179 | . 2 ⊢ ((𝐵 ∩ 𝐴) ∩ 𝐶) = (𝐵 ∩ (𝐴 ∩ 𝐶)) | |
| 5 | 2, 3, 4 | 3eqtr3i 2793 | 1 ⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐵 ∩ (𝐴 ∩ 𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1560 ∩ cin 3903 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-ext 2734 |
| This theorem depends on definitions: df-bi 209 df-an 400 df-tru 1563 df-ex 1800 df-sb 2091 df-clab 2741 df-cleq 2754 df-clel 2837 df-rab 3415 df-v 3456 df-in 3911 |
| This theorem is referenced by: in32 4181 in31 4183 in4 4185 resdmres 6219 kmlem12 10118 ressress 17283 fh1 31818 fh2 31819 mdslmd3i 32532 bj-inrab3 37411 |
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