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Theorem eqsbc3r 3210
Description: eqsbc3 3192 with set variable on right side of equals sign. This proof was automatically generated from the virtual deduction proof eqsbc3rVD 28889 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 2437 . . . . . 6  |-  ( C  =  x  <->  x  =  C )
21sbcbii 3208 . . . . 5  |-  ( [. A  /  x ]. C  =  x  <->  [. A  /  x ]. x  =  C
)
32biimpi 187 . . . 4  |-  ( [. A  /  x ]. C  =  x  ->  [. A  /  x ]. x  =  C )
4 eqsbc3 3192 . . . 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 2437 . . 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 1652    e. wcel 1725   [.wsbc 3153
This theorem is referenced by:  sbcoreleleq  28556  sbcoreleleqVD  28908
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-sbc 3154
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