Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1153 Unicode version

Theorem bnj1153 28502
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1153.1  |-  ( ph  ->  X  e.  A )
bnj1153.2  |-  ( ph  ->  X  e.  B )
Assertion
Ref Expression
bnj1153  |-  ( ph  ->  X  e.  ( A  i^i  B ) )

Proof of Theorem bnj1153
StepHypRef Expression
1 bnj1153.1 . 2  |-  ( ph  ->  X  e.  A )
2 bnj1153.2 . 2  |-  ( ph  ->  X  e.  B )
3 elin 3473 . 2  |-  ( X  e.  ( A  i^i  B )  <->  ( X  e.  A  /\  X  e.  B ) )
41, 2, 3sylanbrc 646 1  |-  ( ph  ->  X  e.  ( A  i^i  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1717    i^i cin 3262
This theorem is referenced by:  bnj1379  28540  bnj1177  28713
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 2368
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 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-v 2901  df-in 3270
  Copyright terms: Public domain W3C validator