![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 3948 | . . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) | |
2 | 1 | ineq1i 3953 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐵 ∩ 𝐴) ∩ 𝐶) |
3 | inass 3966 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵 ∩ 𝐶)) | |
4 | inass 3966 | . 2 ⊢ ((𝐵 ∩ 𝐴) ∩ 𝐶) = (𝐵 ∩ (𝐴 ∩ 𝐶)) | |
5 | 2, 3, 4 | 3eqtr3i 2790 | 1 ⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐵 ∩ (𝐴 ∩ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1632 ∩ cin 3714 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-v 3342 df-in 3722 |
This theorem is referenced by: in32 3968 in31 3970 in4 3972 resdmres 5786 kmlem12 9175 ressress 16140 fh1 28786 fh2 28787 mdslmd3i 29500 bj-inrab3 33231 |
Copyright terms: Public domain | W3C validator |