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Theorem ralimda 41945
 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 3183 . . . 4 𝑥𝑥𝐴 𝜓
31, 2nfan 1900 . . 3 𝑥(𝜑 ∧ ∀𝑥𝐴 𝜓)
4 id 22 . . . . 5 ((𝜑𝑥𝐴) → (𝜑𝑥𝐴))
54adantlr 714 . . . 4 (((𝜑 ∧ ∀𝑥𝐴 𝜓) ∧ 𝑥𝐴) → (𝜑𝑥𝐴))
6 rspa 3171 . . . . 5 ((∀𝑥𝐴 𝜓𝑥𝐴) → 𝜓)
76adantll 713 . . . 4 (((𝜑 ∧ ∀𝑥𝐴 𝜓) ∧ 𝑥𝐴) → 𝜓)
8 ralimda.2 . . . 4 ((𝜑𝑥𝐴) → (𝜓𝜒))
95, 7, 8sylc 65 . . 3 (((𝜑 ∧ ∀𝑥𝐴 𝜓) ∧ 𝑥𝐴) → 𝜒)
103, 9ralrimia 41938 . 2 ((𝜑 ∧ ∀𝑥𝐴 𝜓) → ∀𝑥𝐴 𝜒)
1110ex 416 1 (𝜑 → (∀𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  Ⅎwnf 1785   ∈ wcel 2111  ∀wral 3106 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-12 2175 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-ral 3111 This theorem is referenced by:  xlimmnfvlem1  42642  xlimmnfvlem2  42643  xlimpnfvlem1  42646  xlimpnfvlem2  42647
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