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Theorem splintx 25049
Description: Moving an element out from an intersection. (Contributed by FL, 15-Oct-2012.)
Hypothesis
Ref Expression
splintx.1  |-  ( x  =  B  ->  C  =  D )
Assertion
Ref Expression
splintx  |-  ( B  e.  A  ->  |^|_ x  e.  A  C  =  ( |^|_ x  e.  ( A  \  { B } ) C  i^i  D ) )
Distinct variable groups:    x, A    x, B    x, D
Allowed substitution hint:    C( x)

Proof of Theorem splintx
StepHypRef Expression
1 snssi 3759 . . 3  |-  ( B  e.  A  ->  { B }  C_  A )
2 splint 25048 . . 3  |-  ( { B }  C_  A  -> 
|^|_ x  e.  A  C  =  ( |^|_ x  e.  ( A  \  { B } ) C  i^i  |^|_ x  e.  { B } C ) )
31, 2syl 15 . 2  |-  ( B  e.  A  ->  |^|_ x  e.  A  C  =  ( |^|_ x  e.  ( A  \  { B } ) C  i^i  |^|_
x  e.  { B } C ) )
4 splintx.1 . . . 4  |-  ( x  =  B  ->  C  =  D )
54iinxsng 3978 . . 3  |-  ( B  e.  A  ->  |^|_ x  e.  { B } C  =  D )
65ineq2d 3370 . 2  |-  ( B  e.  A  ->  ( |^|_ x  e.  ( A 
\  { B }
) C  i^i  |^|_ x  e.  { B } C )  =  (
|^|_ x  e.  ( A  \  { B }
) C  i^i  D
) )
73, 6eqtrd 2315 1  |-  ( B  e.  A  ->  |^|_ x  e.  A  C  =  ( |^|_ x  e.  ( A  \  { B } ) C  i^i  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    \ cdif 3149    i^i cin 3151    C_ wss 3152   {csn 3640   |^|_ciin 3906
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  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-sn 3646  df-iin 3908
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