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Theorem 3exdistr 1808
 Description: Distribution of existential quantifiers. (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 900 . . . 4 ((𝜑𝜓𝜒) ↔ (𝜑 ∧ (𝜓𝜒)))
212exbii 1513 . . 3 (∃𝑦𝑧(𝜑𝜓𝜒) ↔ ∃𝑦𝑧(𝜑 ∧ (𝜓𝜒)))
3 19.42vv 1804 . . 3 (∃𝑦𝑧(𝜑 ∧ (𝜓𝜒)) ↔ (𝜑 ∧ ∃𝑦𝑧(𝜓𝜒)))
4 exdistr 1803 . . . 4 (∃𝑦𝑧(𝜓𝜒) ↔ ∃𝑦(𝜓 ∧ ∃𝑧𝜒))
54anbi2i 438 . . 3 ((𝜑 ∧ ∃𝑦𝑧(𝜓𝜒)) ↔ (𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧𝜒)))
62, 3, 53bitri 199 . 2 (∃𝑦𝑧(𝜑𝜓𝜒) ↔ (𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧𝜒)))
76exbii 1512 1 (∃𝑥𝑦𝑧(𝜑𝜓𝜒) ↔ ∃𝑥(𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧𝜒)))
 Colors of variables: wff set class Syntax hints:   ∧ wa 101   ↔ wb 102   ∧ w3a 896  ∃wex 1397 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443 This theorem depends on definitions:  df-bi 114  df-3an 898 This theorem is referenced by: (None)
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