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Theorem alral 2481
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 ax-1 6 . . 3 (𝜑 → (𝑥𝐴𝜑))
21alimi 1432 . 2 (∀𝑥𝜑 → ∀𝑥(𝑥𝐴𝜑))
3 df-ral 2422 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
42, 3sylibr 133 1 (∀𝑥𝜑 → ∀𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1330  wcel 1481  wral 2417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426
This theorem depends on definitions:  df-bi 116  df-ral 2422
This theorem is referenced by:  abnex  4374  find  4519  prodeq2w  11355  findset  13307
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