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Theorem ralimda 39847
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 3090 . . . 4 𝑥𝑥𝐴 𝜓
31, 2nfan 1980 . . 3 𝑥(𝜑 ∧ ∀𝑥𝐴 𝜓)
4 id 22 . . . . 5 ((𝜑𝑥𝐴) → (𝜑𝑥𝐴))
54adantlr 694 . . . 4 (((𝜑 ∧ ∀𝑥𝐴 𝜓) ∧ 𝑥𝐴) → (𝜑𝑥𝐴))
6 rspa 3079 . . . . 5 ((∀𝑥𝐴 𝜓𝑥𝐴) → 𝜓)
76adantll 693 . . . 4 (((𝜑 ∧ ∀𝑥𝐴 𝜓) ∧ 𝑥𝐴) → 𝜓)
8 ralimda.2 . . . 4 ((𝜑𝑥𝐴) → (𝜓𝜒))
95, 7, 8sylc 65 . . 3 (((𝜑 ∧ ∀𝑥𝐴 𝜓) ∧ 𝑥𝐴) → 𝜒)
103, 9ralrimia 39836 . 2 ((𝜑 ∧ ∀𝑥𝐴 𝜓) → ∀𝑥𝐴 𝜒)
1110ex 397 1 (𝜑 → (∀𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wnf 1856  wcel 2145  wral 3061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-10 2174  ax-12 2203
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-ral 3066
This theorem is referenced by:  xlimmnfvlem1  40576  xlimmnfvlem2  40577  xlimpnfvlem1  40580  xlimpnfvlem2  40581
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