MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  3exdistr Structured version   Visualization version   GIF version

Theorem 3exdistr 1955
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 1089 . . . 4 ((𝜑𝜓𝜒) ↔ (𝜑 ∧ (𝜓𝜒)))
212exbii 1842 . . 3 (∃𝑦𝑧(𝜑𝜓𝜒) ↔ ∃𝑦𝑧(𝜑 ∧ (𝜓𝜒)))
3 19.42vv 1951 . . 3 (∃𝑦𝑧(𝜑 ∧ (𝜓𝜒)) ↔ (𝜑 ∧ ∃𝑦𝑧(𝜓𝜒)))
4 exdistr 1948 . . . 4 (∃𝑦𝑧(𝜓𝜒) ↔ ∃𝑦(𝜓 ∧ ∃𝑧𝜒))
54anbi2i 622 . . 3 ((𝜑 ∧ ∃𝑦𝑧(𝜓𝜒)) ↔ (𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧𝜒)))
62, 3, 53bitri 298 . 2 (∃𝑦𝑧(𝜑𝜓𝜒) ↔ (𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧𝜒)))
76exbii 1841 1 (∃𝑥𝑦𝑧(𝜑𝜓𝜒) ↔ ∃𝑥(𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  w3a 1081  wex 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904
This theorem depends on definitions:  df-bi 208  df-an 397  df-3an 1083  df-ex 1774
This theorem is referenced by:  4exdistr  1956
  Copyright terms: Public domain W3C validator