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Theorem brtxpsd3 24436
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 304 . . 3  |-  ( ( x  e.  B  <->  x  e.  X )  <->  ( x  e.  B  <->  x S A ) )
32albii 1553 . 2  |-  ( A. x ( x  e.  B  <->  x  e.  X
)  <->  A. x ( x  e.  B  <->  x S A ) )
4 dfcleq 2277 . 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 24435 . 2  |-  ( A R B  <->  A. x
( x  e.  B  <->  x S A ) )
103, 4, 93bitr4ri 269 1  |-  ( A R B  <->  B  =  X )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   A.wal 1527    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149   class class class wbr 4023    _E cep 4303   ran crn 4690  (++)csymdif 24361    (x) ctxp 24373
This theorem is referenced by:  brbigcup  24438  brsingle  24456  brimage  24465  brcart  24471  brapply  24477  brcup  24478  brcap  24479
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-eprel 4305  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-1st 6122  df-2nd 6123  df-symdif 24362  df-txp 24395
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