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Theorem r19.21 3213
Description: Restricted quantifier version of 19.21 2200. (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 3212 . 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 208  wnf 1778  wral 3136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-12 2170
This theorem depends on definitions:  df-bi 209  df-ex 1775  df-nf 1779  df-ral 3141
This theorem is referenced by:  rmo3f  3723  ra4  3867  rmoanim  3876  rmoanimALT  3877  r19.32  43286
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