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Theorem r19.29 2450
 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 pm3.2 126 . . . 4 (𝜑 → (𝜓 → (𝜑𝜓)))
21ralimi 2384 . . 3 (∀𝑥𝐴 𝜑 → ∀𝑥𝐴 (𝜓 → (𝜑𝜓)))
3 rexim 2413 . . 3 (∀𝑥𝐴 (𝜓 → (𝜑𝜓)) → (∃𝑥𝐴 𝜓 → ∃𝑥𝐴 (𝜑𝜓)))
42, 3syl 14 . 2 (∀𝑥𝐴 𝜑 → (∃𝑥𝐴 𝜓 → ∃𝑥𝐴 (𝜑𝜓)))
54imp 115 1 ((∀𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 𝜓) → ∃𝑥𝐴 (𝜑𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wral 2306  ∃wrex 2307 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427 This theorem depends on definitions:  df-bi 110  df-ral 2311  df-rex 2312 This theorem is referenced by:  r19.29r  2451  r19.29d2r  2455  r19.35-1  2460  triun  3867  ralxfrd  4194  elrnmptg  4586  fun11iun  5147  fmpt  5319  fliftfun  5436  bj-findis  10104
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