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Mirrors > Home > MPE Home > Th. List > ralun | Structured version Visualization version GIF version |
Description: Restricted quantification over union. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
ralun | ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 𝜑) → ∀𝑥 ∈ (𝐴 ∪ 𝐵)𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralunb 4206 | . 2 ⊢ (∀𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 𝜑)) | |
2 | 1 | biimpri 228 | 1 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 𝜑) → ∀𝑥 ∈ (𝐴 ∪ 𝐵)𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∀wral 3058 ∪ cun 3960 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-v 3479 df-un 3967 |
This theorem is referenced by: f1ounsn 7291 ac6sfi 9317 frfi 9318 fpwwe2lem12 10679 modfsummod 15826 drsdirfi 18362 lbsextlem4 21180 fbun 23863 filconn 23906 cnmpopc 24968 chtub 27270 prsiga 34111 finixpnum 37591 poimirlem31 37637 poimirlem32 37638 kelac1 43051 cantnfresb 43313 |
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