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Theorem r19.29 2507
Description: Theorem 19.29 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.) (Proof shortened by Andrew Salmon, 30-May-2011.)
Assertion
Ref Expression
r19.29 ((∀𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 𝜓) → ∃𝑥𝐴 (𝜑𝜓))

Proof of Theorem r19.29
StepHypRef Expression
1 pm3.2 138 . . . 4 (𝜑 → (𝜓 → (𝜑𝜓)))
21ralimi 2439 . . 3 (∀𝑥𝐴 𝜑 → ∀𝑥𝐴 (𝜓 → (𝜑𝜓)))
3 rexim 2468 . . 3 (∀𝑥𝐴 (𝜓 → (𝜑𝜓)) → (∃𝑥𝐴 𝜓 → ∃𝑥𝐴 (𝜑𝜓)))
42, 3syl 14 . 2 (∀𝑥𝐴 𝜑 → (∃𝑥𝐴 𝜓 → ∃𝑥𝐴 (𝜑𝜓)))
54imp 123 1 ((∀𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 𝜓) → ∃𝑥𝐴 (𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wral 2360  wrex 2361
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-4 1446  ax-ial 1473
This theorem depends on definitions:  df-bi 116  df-ral 2365  df-rex 2366
This theorem is referenced by:  r19.29r  2508  r19.29d2r  2513  r19.35-1  2518  triun  3955  ralxfrd  4297  elrnmptg  4700  fun11iun  5287  fmpt  5463  fliftfun  5589  epttop  11844  bj-findis  12140
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