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Theorem 19.29r 1564
Description: Variation of Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)
Assertion
Ref Expression
19.29r ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑𝜓))

Proof of Theorem 19.29r
StepHypRef Expression
1 19.29 1563 . 2 ((∀𝑥𝜓 ∧ ∃𝑥𝜑) → ∃𝑥(𝜓𝜑))
2 ancom 263 . 2 ((∃𝑥𝜑 ∧ ∀𝑥𝜓) ↔ (∀𝑥𝜓 ∧ ∃𝑥𝜑))
3 exancom 1551 . 2 (∃𝑥(𝜑𝜓) ↔ ∃𝑥(𝜓𝜑))
41, 2, 33imtr4i 200 1 ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1294  wex 1433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1388  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-4 1452  ax-ial 1479
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  19.29r2  1565  19.29x  1566  exan  1635  ax9o  1640  equvini  1695  eu2  1999  intab  3739  imadiflem  5127  bj-inex  12506
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