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Mirrors > Home > ILE Home > Th. List > ralun | 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 3299 | . 2 ⊢ (∀𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 𝜑)) | |
2 | 1 | biimpri 132 | 1 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 𝜑) → ∀𝑥 ∈ (𝐴 ∪ 𝐵)𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wral 2442 ∪ cun 3110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-v 2724 df-un 3116 |
This theorem is referenced by: omsinds 4594 ac6sfi 6856 fimax2gtrilemstep 6858 finomni 7096 uzsinds 10368 iseqf1olemqk 10420 seq3f1olemstep 10427 fimaxre2 11158 modfsummod 11389 |
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