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Theorem rexim 3154
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 150 . . . 4 ((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))
21ral2imi 3094 . . 3 (∀𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 ¬ 𝜓 → ∀𝑥𝐴 ¬ 𝜑))
32con3d 149 . 2 (∀𝑥𝐴 (𝜑𝜓) → (¬ ∀𝑥𝐴 ¬ 𝜑 → ¬ ∀𝑥𝐴 ¬ 𝜓))
4 dfrex2 3142 . 2 (∃𝑥𝐴 𝜑 ↔ ¬ ∀𝑥𝐴 ¬ 𝜑)
5 dfrex2 3142 . 2 (∃𝑥𝐴 𝜓 ↔ ¬ ∀𝑥𝐴 ¬ 𝜓)
63, 4, 53imtr4g 287 1 (∀𝑥𝐴 (𝜑𝜓) → (∃𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wral 3055  wrex 3056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904
This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1875  df-ral 3060  df-rex 3061
This theorem is referenced by:  reximia  3155  reximdai  3158  reximdvai  3161  r19.29  3219  reupick2  4077  ss2iun  4692  chfnrn  6518  isf32lem2  9429  ptcmplem4  22138  bnj110  31308  poimirlem25  33790
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