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Theorem brtxpsd3 30711
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 319 . . 3  |-  ( ( x  e.  B  <->  x  e.  X )  <->  ( x  e.  B  <->  x S A ) )
32albii 1701 . 2  |-  ( A. x ( x  e.  B  <->  x  e.  X
)  <->  A. x ( x  e.  B  <->  x S A ) )
4 dfcleq 2455 . 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 30710 . 2  |-  ( A R B  <->  A. x
( x  e.  B  <->  x S A ) )
103, 4, 93bitr4ri 286 1  |-  ( A R B  <->  B  =  X )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189   A.wal 1452    = wceq 1454    e. wcel 1897   _Vcvv 3056    \ cdif 3412    /_\ csymdif 3673   class class class wbr 4415    _E cep 4761   ran crn 4853    (x) ctxp 30644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-symdif 3674  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-br 4416  df-opab 4475  df-mpt 4476  df-eprel 4763  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-fo 5606  df-fv 5608  df-1st 6819  df-2nd 6820  df-txp 30668
This theorem is referenced by:  brbigcup  30713  brsingle  30732  brimage  30741  brcart  30747  brapply  30753  brcup  30754  brcap  30755
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