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Theorem brtxpsd3 25462
Description: A third common abbreviation for quantifier-free definitions. (Contributed by Scott Fenton, 3-May-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
brtxpsd3.5  |-  ( x  e.  X  <->  x S A )
Assertion
Ref Expression
brtxpsd3  |-  ( A R B  <->  B  =  X )
Distinct variable groups:    x, A    x, B    x, S    x, X
Allowed substitution hints:    C( x)    R( x)

Proof of Theorem brtxpsd3
StepHypRef Expression
1 brtxpsd3.5 . . . 4  |-  ( x  e.  X  <->  x S A )
21bibi2i 305 . . 3  |-  ( ( x  e.  B  <->  x  e.  X )  <->  ( x  e.  B  <->  x S A ) )
32albii 1572 . 2  |-  ( A. x ( x  e.  B  <->  x  e.  X
)  <->  A. x ( x  e.  B  <->  x S A ) )
4 dfcleq 2383 . 2  |-  ( B  =  X  <->  A. x
( x  e.  B  <->  x  e.  X ) )
5 brtxpsd2.1 . . 3  |-  A  e. 
_V
6 brtxpsd2.2 . . 3  |-  B  e. 
_V
7 brtxpsd2.3 . . 3  |-  R  =  ( C  \  ran  ( ( _V  (x)  _E  )(++) ( S  (x)  _V ) ) )
8 brtxpsd2.4 . . 3  |-  A C B
95, 6, 7, 8brtxpsd2 25461 . 2  |-  ( A R B  <->  A. x
( x  e.  B  <->  x S A ) )
103, 4, 93bitr4ri 270 1  |-  ( A R B  <->  B  =  X )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   A.wal 1546    = wceq 1649    e. wcel 1717   _Vcvv 2901    \ cdif 3262   class class class wbr 4155    _E cep 4435   ran crn 4821  (++)csymdif 25387    (x) ctxp 25399
This theorem is referenced by:  brbigcup  25464  brsingle  25482  brimage  25491  brcart  25497  brapply  25503  brcup  25504  brcap  25505
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-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-eprel 4437  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-fo 5402  df-fv 5404  df-1st 6290  df-2nd 6291  df-symdif 25388  df-txp 25421
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