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Theorem alral 3077
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 1889 . 2 (∀𝑥𝜑 → ∀𝑥(𝑥𝐴𝜑))
2 df-ral 3066 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
31, 2sylibr 224 1 (∀𝑥𝜑 → ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1629  wcel 2145  wral 3061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885
This theorem depends on definitions:  df-bi 197  df-ral 3066
This theorem is referenced by:  abnex  7112  find  7238  brdom5  9553  brdom4  9554  prodeq2w  14849  rpnnen2lem12  15160  elpotr  32022  phpreu  33726  neik0pk1imk0  38871  ordelordALTVD  39625  rexrsb  41689
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