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Theorem r19.29 2467
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 pm3.2 130 . . . 4 (𝜑 → (𝜓 → (𝜑𝜓)))
21ralimi 2401 . . 3 (∀𝑥𝐴 𝜑 → ∀𝑥𝐴 (𝜓 → (𝜑𝜓)))
3 rexim 2430 . . 3 (∀𝑥𝐴 (𝜓 → (𝜑𝜓)) → (∃𝑥𝐴 𝜓 → ∃𝑥𝐴 (𝜑𝜓)))
42, 3syl 14 . 2 (∀𝑥𝐴 𝜑 → (∃𝑥𝐴 𝜓 → ∃𝑥𝐴 (𝜑𝜓)))
54imp 119 1 ((∀𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 𝜓) → ∃𝑥𝐴 (𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wral 2323  wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-ral 2328  df-rex 2329
This theorem is referenced by:  r19.29r  2468  r19.29d2r  2472  r19.35-1  2477  triun  3895  ralxfrd  4222  elrnmptg  4614  fun11iun  5175  fmpt  5347  fliftfun  5464  bj-findis  10491
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