Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1405 Unicode version

Theorem bnj1405 28539
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj1405.1  |-  ( ph  ->  X  e.  U_ y  e.  A  B )
Assertion
Ref Expression
bnj1405  |-  ( ph  ->  E. y  e.  A  X  e.  B )
Distinct variable group:    y, X
Allowed substitution hints:    ph( y)    A( y)    B( y)

Proof of Theorem bnj1405
StepHypRef Expression
1 bnj1405.1 . 2  |-  ( ph  ->  X  e.  U_ y  e.  A  B )
2 eliun 4032 . 2  |-  ( X  e.  U_ y  e.  A  B  <->  E. y  e.  A  X  e.  B )
31, 2sylib 189 1  |-  ( ph  ->  E. y  e.  A  X  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1717   E.wrex 2643   U_ciun 4028
This theorem is referenced by:  bnj1408  28736  bnj1450  28750  bnj1501  28767
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ral 2647  df-rex 2648  df-v 2894  df-iun 4030
  Copyright terms: Public domain W3C validator