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Theorem alral 2912
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 1755 . 2 (∀𝑥𝜑 → ∀𝑥(𝑥𝐴𝜑))
2 df-ral 2901 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
31, 2sylibr 223 1 (∀𝑥𝜑 → ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1473  wcel 1977  wral 2896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728
This theorem depends on definitions:  df-bi 196  df-ral 2901
This theorem is referenced by:  find  6983  brdom5  9232  brdom4  9233  prodeq2w  14481  rpnnen2lem12  14793  elpotr  30930  phpreu  32563  neik0pk1imk0  37365  ordelordALTVD  38125  rexrsb  39818
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