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Theorem ralrimd 2414
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 1517 . 2 (𝜑 → (𝜓 → ∀𝑥(𝑥𝐴𝜒)))
5 df-ral 2328 . 2 (∀𝑥𝐴 𝜒 ↔ ∀𝑥(𝑥𝐴𝜒))
64, 5syl6ibr 155 1 (𝜑 → (𝜓 → ∀𝑥𝐴 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1257  wnf 1365  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  ax-4 1416
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-ral 2328
This theorem is referenced by:  ralrimdv  2415  fliftfun  5463  fzrevral  9068
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