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Theorem r19.21 2950
 Description: Restricted quantifier version of 19.21 2073. (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 2949 . 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 196  Ⅎwnf 1705  ∀wral 2907 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-12 2044 This theorem depends on definitions:  df-bi 197  df-ex 1702  df-nf 1707  df-ral 2912 This theorem is referenced by:  ra4  3507  rmo3f  29196  rmo4fOLD  29197  r19.32  40487  rmoanim  40499
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