MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  alral Structured version   Visualization version   GIF version

Theorem alral 3097
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 1777 . 2 (∀𝑥𝜑 → ∀𝑥(𝑥𝐴𝜑))
2 df-ral 3086 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
31, 2sylibr 226 1 (∀𝑥𝜑 → ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1506  wcel 2051  wral 3081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773
This theorem depends on definitions:  df-bi 199  df-ral 3086
This theorem is referenced by:  abnex  7294  find  7420  brdom5  9747  brdom4  9748  hashgt23el  13596  prodeq2w  15124  rpnnen2lem12  15436  elpotr  32583  fvineqsnf1  34169  fvineqsneq  34171  phpreu  34354  neik0pk1imk0  39798  ordelordALTVD  40658  rexrsb  42738
  Copyright terms: Public domain W3C validator