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Theorem alral 2911
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 1754 . 2 (∀𝑥𝜑 → ∀𝑥(𝑥𝐴𝜑))
2 df-ral 2900 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
31, 2sylibr 222 1 (∀𝑥𝜑 → ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1472  wcel 1976  wral 2895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727
This theorem depends on definitions:  df-bi 195  df-ral 2900
This theorem is referenced by:  find  6960  brdom5  9209  brdom4  9210  prodeq2w  14429  rpnnen2lem12  14741  elpotr  30723  phpreu  32346  neik0pk1imk0  37148  ordelordALTVD  37908  rexrsb  39601
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