Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-3exbi Structured version   Visualization version   GIF version

Theorem bj-3exbi 33105
Description: Closed form of 3exbii 1946. (Contributed by BJ, 6-May-2019.)
Assertion
Ref Expression
bj-3exbi (∀𝑥𝑦𝑧(𝜑𝜓) → (∃𝑥𝑦𝑧𝜑 ↔ ∃𝑥𝑦𝑧𝜓))

Proof of Theorem bj-3exbi
StepHypRef Expression
1 exbi 1943 . . 3 (∀𝑧(𝜑𝜓) → (∃𝑧𝜑 ↔ ∃𝑧𝜓))
212alimi 1908 . 2 (∀𝑥𝑦𝑧(𝜑𝜓) → ∀𝑥𝑦(∃𝑧𝜑 ↔ ∃𝑧𝜓))
3 bj-2exbi 33104 . 2 (∀𝑥𝑦(∃𝑧𝜑 ↔ ∃𝑧𝜓) → (∃𝑥𝑦𝑧𝜑 ↔ ∃𝑥𝑦𝑧𝜓))
42, 3syl 17 1 (∀𝑥𝑦𝑧(𝜑𝜓) → (∃𝑥𝑦𝑧𝜑 ↔ ∃𝑥𝑦𝑧𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wal 1651  wex 1875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905
This theorem depends on definitions:  df-bi 199  df-ex 1876
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator