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Theorem ralrimd 2451
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 1546 . 2 (𝜑 → (𝜓 → ∀𝑥(𝑥𝐴𝜒)))
5 df-ral 2364 . 2 (∀𝑥𝐴 𝜒 ↔ ∀𝑥(𝑥𝐴𝜒))
64, 5syl6ibr 160 1 (𝜑 → (𝜓 → ∀𝑥𝐴 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1287  wnf 1394  wcel 1438  wral 2359
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-4 1445
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-ral 2364
This theorem is referenced by:  ralrimdv  2452  fliftfun  5575  mapxpen  6562  fzrevral  9515
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