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Theorem 19.29x 1603
Description: Variation of Theorem 19.29 of [Margaris] p. 90 with mixed quantification. (Contributed by NM, 11-Feb-2005.)
Assertion
Ref Expression
19.29x ((∃𝑥𝑦𝜑 ∧ ∀𝑥𝑦𝜓) → ∃𝑥𝑦(𝜑𝜓))

Proof of Theorem 19.29x
StepHypRef Expression
1 19.29r 1601 . 2 ((∃𝑥𝑦𝜑 ∧ ∀𝑥𝑦𝜓) → ∃𝑥(∀𝑦𝜑 ∧ ∃𝑦𝜓))
2 19.29 1600 . . 3 ((∀𝑦𝜑 ∧ ∃𝑦𝜓) → ∃𝑦(𝜑𝜓))
32eximi 1580 . 2 (∃𝑥(∀𝑦𝜑 ∧ ∃𝑦𝜓) → ∃𝑥𝑦(𝜑𝜓))
41, 3syl 14 1 ((∃𝑥𝑦𝜑 ∧ ∀𝑥𝑦𝜓) → ∃𝑥𝑦(𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1330  wex 1469
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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-ial 1515
This theorem depends on definitions:  df-bi 116
This theorem is referenced by: (None)
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