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Theorem ralun 3119
 Description: Restricted quantification over union. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
ralun ((x A φ x B φ) → x (AB)φ)

Proof of Theorem ralun
StepHypRef Expression
1 ralunb 3118 . 2 (x (AB)φ ↔ (x A φ x B φ))
21biimpri 124 1 ((x A φ x B φ) → x (AB)φ)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wral 2300   ∪ cun 2909 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916 This theorem is referenced by: (None)
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