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Theorem ralimralim 38775
Description: Introducing any antecedent in a restricted universal quantification. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Assertion
Ref Expression
ralimralim (∀𝑥𝐴 𝜑 → ∀𝑥𝐴 (𝜓𝜑))

Proof of Theorem ralimralim
StepHypRef Expression
1 nfra1 2937 . 2 𝑥𝑥𝐴 𝜑
2 rspa 2926 . . . 4 ((∀𝑥𝐴 𝜑𝑥𝐴) → 𝜑)
3 ax-1 6 . . . 4 (𝜑 → (𝜓𝜑))
42, 3syl 17 . . 3 ((∀𝑥𝐴 𝜑𝑥𝐴) → (𝜓𝜑))
54ex 450 . 2 (∀𝑥𝐴 𝜑 → (𝑥𝐴 → (𝜓𝜑)))
61, 5ralrimi 2953 1 (∀𝑥𝐴 𝜑 → ∀𝑥𝐴 (𝜓𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  wral 2908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707  df-ral 2913
This theorem is referenced by:  infxrunb2  39083
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