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Theorem r19.21 2939
Description: Restricted quantifier version of 19.21 2062. (Contributed by Scott Fenton, 30-Mar-2011.)
Hypothesis
Ref Expression
r19.21.1 𝑥𝜑
Assertion
Ref Expression
r19.21 (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∀𝑥𝐴 𝜓))

Proof of Theorem r19.21
StepHypRef Expression
1 r19.21.1 . 2 𝑥𝜑
2 r19.21t 2938 . 2 (Ⅎ𝑥𝜑 → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∀𝑥𝐴 𝜓)))
31, 2ax-mp 5 1 (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∀𝑥𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wnf 1699  wral 2896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034
This theorem depends on definitions:  df-bi 196  df-ex 1696  df-nf 1701  df-ral 2901
This theorem is referenced by:  ra4  3491  rmo3f  28719  rmo4fOLD  28720  r19.32  39816  rmoanim  39828
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