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Theorem ralimda 41282
Description: Deduction quantifying both antecedent and consequent. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
ralimda.1 𝑥𝜑
ralimda.2 ((𝜑𝑥𝐴) → (𝜓𝜒))
Assertion
Ref Expression
ralimda (𝜑 → (∀𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜒))

Proof of Theorem ralimda
StepHypRef Expression
1 ralimda.1 . . . 4 𝑥𝜑
2 nfra1 3216 . . . 4 𝑥𝑥𝐴 𝜓
31, 2nfan 1891 . . 3 𝑥(𝜑 ∧ ∀𝑥𝐴 𝜓)
4 id 22 . . . . 5 ((𝜑𝑥𝐴) → (𝜑𝑥𝐴))
54adantlr 711 . . . 4 (((𝜑 ∧ ∀𝑥𝐴 𝜓) ∧ 𝑥𝐴) → (𝜑𝑥𝐴))
6 rspa 3203 . . . . 5 ((∀𝑥𝐴 𝜓𝑥𝐴) → 𝜓)
76adantll 710 . . . 4 (((𝜑 ∧ ∀𝑥𝐴 𝜓) ∧ 𝑥𝐴) → 𝜓)
8 ralimda.2 . . . 4 ((𝜑𝑥𝐴) → (𝜓𝜒))
95, 7, 8sylc 65 . . 3 (((𝜑 ∧ ∀𝑥𝐴 𝜓) ∧ 𝑥𝐴) → 𝜒)
103, 9ralrimia 41274 . 2 ((𝜑 ∧ ∀𝑥𝐴 𝜓) → ∀𝑥𝐴 𝜒)
1110ex 413 1 (𝜑 → (∀𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wnf 1775  wcel 2105  wral 3135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-10 2136  ax-12 2167
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-ral 3140
This theorem is referenced by:  xlimmnfvlem1  41989  xlimmnfvlem2  41990  xlimpnfvlem1  41993  xlimpnfvlem2  41994
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