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Theorem ra4 3720
Description: Restricted quantifier version of Axiom 5 of [Mendelson] p. 69. This is the axiom stdpc5 2242 of standard predicate calculus for a restricted domain. See ra4v 3719 for a version requiring fewer axioms. (Contributed by NM, 16-Jan-2004.) (Proof shortened by BJ, 27-Mar-2020.)
Hypothesis
Ref Expression
ra4.1 𝑥𝜑
Assertion
Ref Expression
ra4 (∀𝑥𝐴 (𝜑𝜓) → (𝜑 → ∀𝑥𝐴 𝜓))

Proof of Theorem ra4
StepHypRef Expression
1 ra4.1 . . 3 𝑥𝜑
21r19.21 3137 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∀𝑥𝐴 𝜓))
32biimpi 208 1 (∀𝑥𝐴 (𝜑𝜓) → (𝜑 → ∀𝑥𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wnf 1879  wral 3089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-12 2213
This theorem depends on definitions:  df-bi 199  df-ex 1876  df-nf 1880  df-ral 3094
This theorem is referenced by: (None)
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