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Theorem 19.29 1584
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 pm3.2 138 . . . 4 (𝜑 → (𝜓 → (𝜑𝜓)))
21alimi 1416 . . 3 (∀𝑥𝜑 → ∀𝑥(𝜓 → (𝜑𝜓)))
3 exim 1563 . . 3 (∀𝑥(𝜓 → (𝜑𝜓)) → (∃𝑥𝜓 → ∃𝑥(𝜑𝜓)))
42, 3syl 14 . 2 (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥(𝜑𝜓)))
54imp 123 1 ((∀𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1314  wex 1453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-ial 1499
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  19.29r  1585  19.29x  1587  19.35-1  1588  equs4  1688  equvini  1716  rexxfrd  4354  funimaexglem  5176  bj-inex  13032
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