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Theorem alral 3066
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 1890 . 2 (∀𝑥𝜑 → ∀𝑥(𝑥𝐴𝜑))
2 df-ral 3055 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
31, 2sylibr 224 1 (∀𝑥𝜑 → ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1630  wcel 2139  wral 3050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886
This theorem depends on definitions:  df-bi 197  df-ral 3055
This theorem is referenced by:  abnex  7131  find  7257  brdom5  9563  brdom4  9564  prodeq2w  14861  rpnnen2lem12  15173  elpotr  32012  phpreu  33724  neik0pk1imk0  38865  ordelordALTVD  39620  rexrsb  41693
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