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Theorem 19.29 1511
 Description: Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
Assertion
Ref Expression
19.29 ((∀𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))

Proof of Theorem 19.29
StepHypRef Expression
1 pm3.2 126 . . . 4 (𝜑 → (𝜓 → (𝜑𝜓)))
21alimi 1344 . . 3 (∀𝑥𝜑 → ∀𝑥(𝜓 → (𝜑𝜓)))
3 exim 1490 . . 3 (∀𝑥(𝜓 → (𝜑𝜓)) → (∃𝑥𝜓 → ∃𝑥(𝜑𝜓)))
42, 3syl 14 . 2 (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥(𝜑𝜓)))
54imp 115 1 ((∀𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1241  ∃wex 1381 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 This theorem is referenced by:  19.29r  1512  19.29x  1514  19.35-1  1515  equs4  1613  equvini  1641  rexxfrd  4195  funimaexglem  4982  bj-inex  10027
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