Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  19.42vvv GIF version

Theorem 19.42vvv 1892
 Description: Theorem 19.42 of [Margaris] p. 90 with 3 quantifiers. (Contributed by NM, 21-Sep-2011.)
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 19.42vv 1891 . . 3 (∃𝑦𝑧(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑦𝑧𝜓))
21exbii 1585 . 2 (∃𝑥𝑦𝑧(𝜑𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝑧𝜓))
3 19.42v 1886 . 2 (∃𝑥(𝜑 ∧ ∃𝑦𝑧𝜓) ↔ (𝜑 ∧ ∃𝑥𝑦𝑧𝜓))
42, 3bitri 183 1 (∃𝑥𝑦𝑧(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝑦𝑧𝜓))
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ↔ wb 104  ∃wex 1472 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-17 1506  ax-ial 1514 This theorem depends on definitions:  df-bi 116 This theorem is referenced by:  ceqsex6v  2756
 Copyright terms: Public domain W3C validator