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Theorem eqsbc3r 3161
Description: eqsbc3 3143 with set variable on right side of equals sign. This proof was automatically generated from the virtual deduction proof eqsbc3rVD 28293 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.)
Assertion
Ref Expression
eqsbc3r  |-  ( A  e.  B  ->  ( [. A  /  x ]. C  =  x  <->  C  =  A ) )
Distinct variable groups:    x, C    x, A
Allowed substitution hint:    B( x)

Proof of Theorem eqsbc3r
StepHypRef Expression
1 eqcom 2389 . . . . . 6  |-  ( C  =  x  <->  x  =  C )
21sbcbii 3159 . . . . 5  |-  ( [. A  /  x ]. C  =  x  <->  [. A  /  x ]. x  =  C
)
32biimpi 187 . . . 4  |-  ( [. A  /  x ]. C  =  x  ->  [. A  /  x ]. x  =  C )
4 eqsbc3 3143 . . . 4  |-  ( A  e.  B  ->  ( [. A  /  x ]. x  =  C  <->  A  =  C ) )
53, 4syl5ib 211 . . 3  |-  ( A  e.  B  ->  ( [. A  /  x ]. C  =  x  ->  A  =  C ) )
6 eqcom 2389 . . 3  |-  ( A  =  C  <->  C  =  A )
75, 6syl6ib 218 . 2  |-  ( A  e.  B  ->  ( [. A  /  x ]. C  =  x  ->  C  =  A ) )
8 idd 22 . . . . 5  |-  ( A  e.  B  ->  ( C  =  A  ->  C  =  A ) )
98, 6syl6ibr 219 . . . 4  |-  ( A  e.  B  ->  ( C  =  A  ->  A  =  C ) )
109, 4sylibrd 226 . . 3  |-  ( A  e.  B  ->  ( C  =  A  ->  [. A  /  x ]. x  =  C )
)
1110, 2syl6ibr 219 . 2  |-  ( A  e.  B  ->  ( C  =  A  ->  [. A  /  x ]. C  =  x )
)
127, 11impbid 184 1  |-  ( A  e.  B  ->  ( [. A  /  x ]. C  =  x  <->  C  =  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1717   [.wsbc 3104
This theorem is referenced by:  sbcoreleleq  27962  sbcoreleleqVD  28312
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-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-v 2901  df-sbc 3105
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