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Theorem r19.23 3233
Description: Restricted quantifier version of 19.23 2209. See r19.23v 3198 for a version requiring fewer axioms. (Contributed by NM, 22-Oct-2010.) (Proof shortened by Mario Carneiro, 8-Oct-2016.)
Hypothesis
Ref Expression
r19.23.1 𝑥𝜓
Assertion
Ref Expression
r19.23 (∀𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓))

Proof of Theorem r19.23
StepHypRef Expression
1 r19.23.1 . 2 𝑥𝜓
2 r19.23t 3232 . 2 (Ⅎ𝑥𝜓 → (∀𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓)))
31, 2ax-mp 5 1 (∀𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wnf 1791  wral 3061  wrex 3062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-12 2175
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-nf 1792  df-ral 3066  df-rex 3067
This theorem is referenced by:  rexlimi  3234  iunssf  4953  ralxp3f  33401  ss2iundf  40944
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