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Theorem 19.42vvv 1961
 Description: Version of 19.42 2239 with three quantifiers and a disjoint variable condition requiring fewer axioms. (Contributed by NM, 21-Sep-2011.) (Proof shortened by Wolf Lammen, 27-Aug-2023.)
Assertion
Ref Expression
19.42vvv (∃𝑥𝑦𝑧(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝑦𝑧𝜓))
Distinct variable groups:   𝜑,𝑥   𝜑,𝑦   𝜑,𝑧
Allowed substitution hints:   𝜓(𝑥,𝑦,𝑧)

Proof of Theorem 19.42vvv
StepHypRef Expression
1 exdistr2 1960 . 2 (∃𝑥𝑦𝑧(𝜑𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝑧𝜓))
2 19.42v 1955 . 2 (∃𝑥(𝜑 ∧ ∃𝑦𝑧𝜓) ↔ (𝜑 ∧ ∃𝑥𝑦𝑧𝜓))
31, 2bitri 278 1 (∃𝑥𝑦𝑧(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝑦𝑧𝜓))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399  ∃wex 1781 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782 This theorem is referenced by:  ceqsex6v  3524
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