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| Mirrors > Home > MPE Home > Th. List > in4 | Structured version Visualization version GIF version | ||
| Description: Rearrangement of intersection of 4 classes. (Contributed by NM, 21-Apr-2001.) |
| Ref | Expression |
|---|---|
| in4 | ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 ∩ 𝐷)) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐷)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | in12 4180 | . . 3 ⊢ (𝐵 ∩ (𝐶 ∩ 𝐷)) = (𝐶 ∩ (𝐵 ∩ 𝐷)) | |
| 2 | 1 | ineq2i 4169 | . 2 ⊢ (𝐴 ∩ (𝐵 ∩ (𝐶 ∩ 𝐷))) = (𝐴 ∩ (𝐶 ∩ (𝐵 ∩ 𝐷))) |
| 3 | inass 4179 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 ∩ 𝐷)) = (𝐴 ∩ (𝐵 ∩ (𝐶 ∩ 𝐷))) | |
| 4 | inass 4179 | . 2 ⊢ ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐷)) = (𝐴 ∩ (𝐶 ∩ (𝐵 ∩ 𝐷))) | |
| 5 | 2, 3, 4 | 3eqtr4i 2794 | 1 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 ∩ 𝐷)) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐷)) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1568 ∩ cin 3903 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-ext 2733 |
| This theorem depends on definitions: df-bi 210 df-an 401 df-tru 1571 df-ex 1808 df-sb 2095 df-clab 2740 df-cleq 2753 df-clel 2836 df-rab 3415 df-v 3455 df-in 3911 |
| This theorem is referenced by: inindi 4186 inindir 4187 fh2 31937 disjxpin 32899 |
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