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Theorem 3exdistr 2035
 Description: Distribution of existential quantifiers in a triple conjunction. (Contributed by NM, 9-Mar-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Assertion
Ref Expression
3exdistr (∃𝑥𝑦𝑧(𝜑𝜓𝜒) ↔ ∃𝑥(𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧𝜒)))
Distinct variable groups:   𝜑,𝑦   𝜑,𝑧   𝜓,𝑧
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦,𝑧)

Proof of Theorem 3exdistr
StepHypRef Expression
1 3anass 1081 . . . 4 ((𝜑𝜓𝜒) ↔ (𝜑 ∧ (𝜓𝜒)))
212exbii 1924 . . 3 (∃𝑦𝑧(𝜑𝜓𝜒) ↔ ∃𝑦𝑧(𝜑 ∧ (𝜓𝜒)))
3 19.42vv 2032 . . 3 (∃𝑦𝑧(𝜑 ∧ (𝜓𝜒)) ↔ (𝜑 ∧ ∃𝑦𝑧(𝜓𝜒)))
4 exdistr 2031 . . . 4 (∃𝑦𝑧(𝜓𝜒) ↔ ∃𝑦(𝜓 ∧ ∃𝑧𝜒))
54anbi2i 732 . . 3 ((𝜑 ∧ ∃𝑦𝑧(𝜓𝜒)) ↔ (𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧𝜒)))
62, 3, 53bitri 286 . 2 (∃𝑦𝑧(𝜑𝜓𝜒) ↔ (𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧𝜒)))
76exbii 1923 1 (∃𝑥𝑦𝑧(𝜑𝜓𝜒) ↔ ∃𝑥(𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧𝜒)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 383   ∧ w3a 1072  ∃wex 1853 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988 This theorem depends on definitions:  df-bi 197  df-an 385  df-3an 1074  df-ex 1854 This theorem is referenced by:  4exdistr  2036
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