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Theorem brintclab 13120
Description: Two ways to express a binary relation which is the intersection of a class. (Contributed by RP, 4-Apr-2020.)
Assertion
Ref Expression
brintclab  |-  ( A
|^| { x  |  ph } B  <->  A. x ( ph  -> 
<. A ,  B >.  e.  x ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    ph( x)

Proof of Theorem brintclab
StepHypRef Expression
1 df-br 4419 . 2  |-  ( A
|^| { x  |  ph } B  <->  <. A ,  B >.  e.  |^| { x  | 
ph } )
2 opex 4681 . . 3  |-  <. A ,  B >.  e.  _V
32elintab 4259 . 2  |-  ( <. A ,  B >.  e. 
|^| { x  |  ph } 
<-> 
A. x ( ph  -> 
<. A ,  B >.  e.  x ) )
41, 3bitri 257 1  |-  ( A
|^| { x  |  ph } B  <->  A. x ( ph  -> 
<. A ,  B >.  e.  x ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   A.wal 1453    e. wcel 1898   {cab 2448   <.cop 3986   |^|cint 4248   class class class wbr 4418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pr 4656
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-int 4249  df-br 4419
This theorem is referenced by:  brtrclfv  13121
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