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

Theorem 3exbii 1766
 Description: Inference adding three existential quantifiers to both sides of an equivalence. (Contributed by NM, 2-May-1995.)
Hypothesis
Ref Expression
3exbii.1 (𝜑𝜓)
Assertion
Ref Expression
3exbii (∃𝑥𝑦𝑧𝜑 ↔ ∃𝑥𝑦𝑧𝜓)

Proof of Theorem 3exbii
StepHypRef Expression
1 3exbii.1 . . 3 (𝜑𝜓)
21exbii 1764 . 2 (∃𝑧𝜑 ↔ ∃𝑧𝜓)
322exbii 1765 1 (∃𝑥𝑦𝑧𝜑 ↔ ∃𝑥𝑦𝑧𝜓)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195  ∃wex 1695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728 This theorem depends on definitions:  df-bi 196  df-ex 1696 This theorem is referenced by:  4exdistr  1911  ceqsex6v  3221  oprabid  6576  dfoprab2  6599  dftpos3  7257  xpassen  7939  bnj916  30257  bnj917  30258  bnj983  30275  bnj996  30279  bnj1021  30288  bnj1033  30291  ellines  31429
 Copyright terms: Public domain W3C validator