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Theorem r19.21v 2554
Description: Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
Assertion
Ref Expression
r19.21v (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∀𝑥𝐴 𝜓))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem r19.21v
StepHypRef Expression
1 nfv 1528 . 2 𝑥𝜑
21r19.21 2553 1 (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∀𝑥𝐴 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wral 2455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-4 1510  ax-17 1526  ax-ial 1534  ax-i5r 1535
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-ral 2460
This theorem is referenced by:  r19.32vdc  2626  rmo4  2930  rmo3  3054  dftr5  4104  reusv3  4460  tfrlem1  6308  tfrlemi1  6332  tfr1onlemaccex  6348  tfrcllemaccex  6361  tfri3  6367  ordiso2  7033  raluz2  9578  ndvdssub  11934  nninfalllem1  14727  nninfsellemqall  14734
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