MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  19.41vvv Structured version   Visualization version   GIF version

Theorem 19.41vvv 1913
Description: Version of 19.41 2101 with three quantifiers and a dv condition requiring fewer axioms. (Contributed by NM, 30-Apr-1995.)
Assertion
Ref Expression
19.41vvv (∃𝑥𝑦𝑧(𝜑𝜓) ↔ (∃𝑥𝑦𝑧𝜑𝜓))
Distinct variable groups:   𝜓,𝑥   𝜓,𝑦   𝜓,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem 19.41vvv
StepHypRef Expression
1 19.41vv 1912 . . 3 (∃𝑦𝑧(𝜑𝜓) ↔ (∃𝑦𝑧𝜑𝜓))
21exbii 1771 . 2 (∃𝑥𝑦𝑧(𝜑𝜓) ↔ ∃𝑥(∃𝑦𝑧𝜑𝜓))
3 19.41v 1911 . 2 (∃𝑥(∃𝑦𝑧𝜑𝜓) ↔ (∃𝑥𝑦𝑧𝜑𝜓))
42, 3bitri 264 1 (∃𝑥𝑦𝑧(𝜑𝜓) ↔ (∃𝑥𝑦𝑧𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  wex 1701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702
This theorem is referenced by:  19.41vvvv  1914  eloprabga  6712  dftpos3  7330
  Copyright terms: Public domain W3C validator