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Theorem ralrimd 2942
Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 16-Feb-2004.)
Hypotheses
Ref Expression
ralrimd.1 𝑥𝜑
ralrimd.2 𝑥𝜓
ralrimd.3 (𝜑 → (𝜓 → (𝑥𝐴𝜒)))
Assertion
Ref Expression
ralrimd (𝜑 → (𝜓 → ∀𝑥𝐴 𝜒))

Proof of Theorem ralrimd
StepHypRef Expression
1 ralrimd.1 . . 3 𝑥𝜑
2 ralrimd.2 . . 3 𝑥𝜓
3 ralrimd.3 . . 3 (𝜑 → (𝜓 → (𝑥𝐴𝜒)))
41, 2, 3alrimd 2071 . 2 (𝜑 → (𝜓 → ∀𝑥(𝑥𝐴𝜒)))
5 df-ral 2901 . 2 (∀𝑥𝐴 𝜒 ↔ ∀𝑥(𝑥𝐴𝜒))
64, 5syl6ibr 241 1 (𝜑 → (𝜓 → ∀𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1473  wnf 1699  wcel 1977  wral 2896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034
This theorem depends on definitions:  df-bi 196  df-ex 1696  df-nf 1701  df-ral 2901
This theorem is referenced by:  reusv2lem3  4792  fliftfun  6440  mapxpen  7989  domtriomlem  9125  dedekind  10052  fzrevral  12252  matunitlindflem2  32370  riotasv3d  33058  ssralv2  37552
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