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

Theorem 19.42vvvv 1906
Description: Theorem 19.42 of [Margaris] p. 90 with 4 quantifiers. (Contributed by Jim Kingdon, 23-Nov-2019.)
Assertion
Ref Expression
19.42vvvv (∃𝑤𝑥𝑦𝑧(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑤𝑥𝑦𝑧𝜓))
Distinct variable groups:   𝜑,𝑤   𝜑,𝑥   𝜑,𝑦   𝜑,𝑧
Allowed substitution hints:   𝜓(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem 19.42vvvv
StepHypRef Expression
1 19.42vv 1904 . . 3 (∃𝑦𝑧(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑦𝑧𝜓))
212exbii 1599 . 2 (∃𝑤𝑥𝑦𝑧(𝜑𝜓) ↔ ∃𝑤𝑥(𝜑 ∧ ∃𝑦𝑧𝜓))
3 19.42vv 1904 . 2 (∃𝑤𝑥(𝜑 ∧ ∃𝑦𝑧𝜓) ↔ (𝜑 ∧ ∃𝑤𝑥𝑦𝑧𝜓))
42, 3bitri 183 1 (∃𝑤𝑥𝑦𝑧(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑤𝑥𝑦𝑧𝜓))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wex 1485
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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  ceqsex8v  2775  enq0tr  7383
  Copyright terms: Public domain W3C validator