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Theorem 19.29 1527
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 pm3.2 130 . . . 4 (𝜑 → (𝜓 → (𝜑𝜓)))
21alimi 1360 . . 3 (∀𝑥𝜑 → ∀𝑥(𝜓 → (𝜑𝜓)))
3 exim 1506 . . 3 (∀𝑥(𝜓 → (𝜑𝜓)) → (∃𝑥𝜓 → ∃𝑥(𝜑𝜓)))
42, 3syl 14 . 2 (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥(𝜑𝜓)))
54imp 119 1 ((∀𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wal 1257  wex 1397
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
This theorem is referenced by:  19.29r  1528  19.29x  1530  19.35-1  1531  equs4  1629  equvini  1657  rexxfrd  4223  funimaexglem  5010  bj-inex  10414
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