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Theorem 19.29x 1868
Description: Variation of 19.29 1865 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 1866 . 2 ((∃𝑥𝑦𝜑 ∧ ∀𝑥𝑦𝜓) → ∃𝑥(∀𝑦𝜑 ∧ ∃𝑦𝜓))
2 19.29 1865 . . 3 ((∀𝑦𝜑 ∧ ∃𝑦𝜓) → ∃𝑦(𝜑𝜓))
32eximi 1826 . 2 (∃𝑥(∀𝑦𝜑 ∧ ∃𝑦𝜓) → ∃𝑥𝑦(𝜑𝜓))
41, 3syl 17 1 ((∃𝑥𝑦𝜑 ∧ ∀𝑥𝑦𝜓) → ∃𝑥𝑦(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1526  wex 1771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772
This theorem is referenced by: (None)
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