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Theorem rexim 3241
Description: Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Nov-1994.) (Proof shortened by Andrew Salmon, 30-May-2011.)
Assertion
Ref Expression
rexim (∀𝑥𝐴 (𝜑𝜓) → (∃𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))

Proof of Theorem rexim
StepHypRef Expression
1 con3 156 . . . 4 ((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))
21ral2imi 3156 . . 3 (∀𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 ¬ 𝜓 → ∀𝑥𝐴 ¬ 𝜑))
32con3d 155 . 2 (∀𝑥𝐴 (𝜑𝜓) → (¬ ∀𝑥𝐴 ¬ 𝜑 → ¬ ∀𝑥𝐴 ¬ 𝜓))
4 dfrex2 3239 . 2 (∃𝑥𝐴 𝜑 ↔ ¬ ∀𝑥𝐴 ¬ 𝜑)
5 dfrex2 3239 . 2 (∃𝑥𝐴 𝜓 ↔ ¬ ∀𝑥𝐴 ¬ 𝜓)
63, 4, 53imtr4g 298 1 (∀𝑥𝐴 (𝜑𝜓) → (∃𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wral 3138  wrex 3139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-ral 3143  df-rex 3144
This theorem is referenced by:  reximia  3242  r19.29  3254  reximdvai  3272  reximdai  3311  r19.35  3341  reupick2  4288  ss2iun  4936  chfnrn  6818  isf32lem2  9775  ptcmplem4  22662  bnj110  32130  poimirlem25  34916
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