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Theorem alral 2384
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 5 . . 3 (𝜑 → (𝑥𝐴𝜑))
21alimi 1360 . 2 (∀𝑥𝜑 → ∀𝑥(𝑥𝐴𝜑))
3 df-ral 2328 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
42, 3sylibr 141 1 (∀𝑥𝜑 → ∀𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1257  wcel 1409  wral 2323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354
This theorem depends on definitions:  df-bi 114  df-ral 2328
This theorem is referenced by:  find  4350  findset  10457
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