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Theorem ralimralim 40177
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 3122 . 2 𝑥𝑥𝐴 𝜑
2 rspa 3111 . . . 4 ((∀𝑥𝐴 𝜑𝑥𝐴) → 𝜑)
3 ax-1 6 . . . 4 (𝜑 → (𝜓𝜑))
42, 3syl 17 . . 3 ((∀𝑥𝐴 𝜑𝑥𝐴) → (𝜓𝜑))
54ex 403 . 2 (∀𝑥𝐴 𝜑 → (𝑥𝐴 → (𝜓𝜑)))
61, 5ralrimi 3138 1 (∀𝑥𝐴 𝜑 → ∀𝑥𝐴 (𝜓𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wcel 2106  wral 3089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-10 2134  ax-12 2162
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-ex 1824  df-nf 1828  df-ral 3094
This theorem is referenced by:  infxrunb2  40485
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