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Theorem brtxpsd2 25742
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 4220 . . . 4  |-  ( A R B  <->  A ( C  \  ran  ( ( _V  (x)  _E  )(++) ( S  (x)  _V )
) ) B )
4 brdif 4262 . . . 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 242 . . 3  |-  ( A R B  <->  ( A C B  /\  -.  A ran  ( ( _V  (x)  _E  )(++) ( S  (x)  _V ) ) B ) )
61, 5mpbiran 886 . 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 25741 . 2  |-  ( -.  A ran  ( ( _V  (x)  _E  )(++) ( S  (x)  _V )
) B  <->  A. x
( x  e.  B  <->  x S A ) )
106, 9bitri 242 1  |-  ( A R B  <->  A. x
( x  e.  B  <->  x S A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 178    /\ wa 360   A.wal 1550    = wceq 1653    e. wcel 1726   _Vcvv 2958    \ cdif 3319   class class class wbr 4214    _E cep 4494   ran crn 4881  (++)csymdif 25664    (x) ctxp 25676
This theorem is referenced by:  brtxpsd3  25743
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-eprel 4496  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fo 5462  df-fv 5464  df-1st 6351  df-2nd 6352  df-symdif 25665  df-txp 25700
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