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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iniin2 | Structured version Visualization version GIF version |
Description: Indexed intersection of intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
iniin2 | ⊢ (𝐴 ≠ ∅ → (𝐵 ∩ ∩ 𝑥 ∈ 𝐴 𝐶) = ∩ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iinin2 4823 | . 2 ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (𝐵 ∩ ∩ 𝑥 ∈ 𝐴 𝐶)) | |
2 | 1 | eqcomd 2783 | 1 ⊢ (𝐴 ≠ ∅ → (𝐵 ∩ ∩ 𝑥 ∈ 𝐴 𝐶) = ∩ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ≠ wne 2968 ∩ cin 3790 ∅c0 4140 ∩ ciin 4754 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-ext 2753 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-v 3399 df-dif 3794 df-in 3798 df-nul 4141 df-iin 4756 |
This theorem is referenced by: (None) |
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