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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 3756 | . 2 ⊢ (∀𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 𝜑)) | |
2 | 1 | biimpri 217 | 1 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 𝜑) → ∀𝑥 ∈ (𝐴 ∪ 𝐵)𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∀wral 2896 ∪ cun 3538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-v 3175 df-un 3545 |
This theorem is referenced by: ac6sfi 8089 frfi 8090 fpwwe2lem13 9343 modfsummod 14367 drsdirfi 16761 lbsextlem4 18982 fbun 21454 filcon 21497 cnmpt2pc 22535 chtub 24737 eupap1 26503 prsiga 29521 finixpnum 32564 poimirlem31 32610 poimirlem32 32611 kelac1 36651 |
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