Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brtxpsd2 Unicode version

Theorem brtxpsd2 24506
Description: Another common abbreviation for quantifier-free definitions. (Contributed by Scott Fenton, 21-Apr-2014.)
Hypotheses
Ref Expression
brtxpsd2.1  |-  A  e. 
_V
brtxpsd2.2  |-  B  e. 
_V
brtxpsd2.3  |-  R  =  ( C  \  ran  ( ( _V  (x)  _E  )(++) ( S  (x)  _V ) ) )
brtxpsd2.4  |-  A C B
Assertion
Ref Expression
brtxpsd2  |-  ( A R B  <->  A. x
( x  e.  B  <->  x S A ) )
Distinct variable groups:    x, A    x, B    x, S
Allowed substitution hints:    C( x)    R( x)

Proof of Theorem brtxpsd2
StepHypRef Expression
1 brtxpsd2.4 . . 3  |-  A C B
2 brtxpsd2.3 . . . . 5  |-  R  =  ( C  \  ran  ( ( _V  (x)  _E  )(++) ( S  (x)  _V ) ) )
32breqi 4045 . . . 4  |-  ( A R B  <->  A ( C  \  ran  ( ( _V  (x)  _E  )(++) ( S  (x)  _V )
) ) B )
4 brdif 4087 . . . 4  |-  ( A ( C  \  ran  ( ( _V  (x)  _E  )(++) ( S  (x)  _V ) ) ) B  <-> 
( A C B  /\  -.  A ran  ( ( _V  (x)  _E  )(++) ( S  (x)  _V ) ) B ) )
53, 4bitri 240 . . 3  |-  ( A R B  <->  ( A C B  /\  -.  A ran  ( ( _V  (x)  _E  )(++) ( S  (x)  _V ) ) B ) )
61, 5mpbiran 884 . 2  |-  ( A R B  <->  -.  A ran  ( ( _V  (x)  _E  )(++) ( S  (x)  _V ) ) B )
7 brtxpsd2.1 . . 3  |-  A  e. 
_V
8 brtxpsd2.2 . . 3  |-  B  e. 
_V
97, 8brtxpsd 24505 . 2  |-  ( -.  A ran  ( ( _V  (x)  _E  )(++) ( S  (x)  _V )
) B  <->  A. x
( x  e.  B  <->  x S A ) )
106, 9bitri 240 1  |-  ( A R B  <->  A. x
( x  e.  B  <->  x S A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ wa 358   A.wal 1530    = wceq 1632    e. wcel 1696   _Vcvv 2801    \ cdif 3162   class class class wbr 4039    _E cep 4319   ran crn 4706  (++)csymdif 24432    (x) ctxp 24444
This theorem is referenced by:  brtxpsd3  24507
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-eprel 4321  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-1st 6138  df-2nd 6139  df-symdif 24433  df-txp 24466
  Copyright terms: Public domain W3C validator