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Theorem 19.29 1556
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 pm3.2 137 . . . 4 (𝜑 → (𝜓 → (𝜑𝜓)))
21alimi 1389 . . 3 (∀𝑥𝜑 → ∀𝑥(𝜓 → (𝜑𝜓)))
3 exim 1535 . . 3 (∀𝑥(𝜓 → (𝜑𝜓)) → (∃𝑥𝜓 → ∃𝑥(𝜑𝜓)))
42, 3syl 14 . 2 (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥(𝜑𝜓)))
54imp 122 1 ((∀𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wal 1287  wex 1426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-ial 1472
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  19.29r  1557  19.29x  1559  19.35-1  1560  equs4  1660  equvini  1688  rexxfrd  4285  funimaexglem  5097  bj-inex  11753
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