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Theorem alral 3075
Description: Universal quantification implies restricted quantification. (Contributed by NM, 20-Oct-2006.)
Assertion
Ref Expression
alral (∀𝑥𝜑 → ∀𝑥𝐴 𝜑)

Proof of Theorem alral
StepHypRef Expression
1 ala1 1815 . 2 (∀𝑥𝜑 → ∀𝑥(𝑥𝐴𝜑))
2 df-ral 3062 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
31, 2sylibr 233 1 (∀𝑥𝜑 → ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1539  wcel 2106  wral 3061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811
This theorem depends on definitions:  df-bi 206  df-ral 3062
This theorem is referenced by:  abnex  7740  find  7883  findOLD  7884  brdom5  10520  brdom4  10521  hashgt23el  14380  prodeq2w  15852  rpnnen2lem12  16164  umgr2cycllem  34119  umgr2cycl  34120  elpotr  34741  fvineqsnf1  36279  fvineqsneq  36281  phpreu  36460  ordelordALTVD  43613  rexrsb  45794
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