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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 3331 | . 2 ⊢ (∀𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 𝜑)) | |
2 | 1 | biimpri 133 | 1 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 𝜑) → ∀𝑥 ∈ (𝐴 ∪ 𝐵)𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∀wral 2468 ∪ cun 3142 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-v 2754 df-un 3148 |
This theorem is referenced by: omsinds 4636 ac6sfi 6916 fimax2gtrilemstep 6918 finomni 7156 uzsinds 10460 iseqf1olemqk 10512 seq3f1olemstep 10519 fimaxre2 11253 modfsummod 11484 |
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