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Theorem eqsbc3r 3048
Description: eqsbc3 3030 with set variable on right side of equals sign. This proof was automatically generated from the virtual deduction proof eqsbc3rVD 28616 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 2285 . . . . . 6  |-  ( C  =  x  <->  x  =  C )
21sbcbii 3046 . . . . 5  |-  ( [. A  /  x ]. C  =  x  <->  [. A  /  x ]. x  =  C
)
32biimpi 186 . . . 4  |-  ( [. A  /  x ]. C  =  x  ->  [. A  /  x ]. x  =  C )
4 eqsbc3 3030 . . . 4  |-  ( A  e.  B  ->  ( [. A  /  x ]. x  =  C  <->  A  =  C ) )
53, 4syl5ib 210 . . 3  |-  ( A  e.  B  ->  ( [. A  /  x ]. C  =  x  ->  A  =  C ) )
6 eqcom 2285 . . 3  |-  ( A  =  C  <->  C  =  A )
75, 6syl6ib 217 . 2  |-  ( A  e.  B  ->  ( [. A  /  x ]. C  =  x  ->  C  =  A ) )
8 idd 21 . . . . 5  |-  ( A  e.  B  ->  ( C  =  A  ->  C  =  A ) )
98, 6syl6ibr 218 . . . 4  |-  ( A  e.  B  ->  ( C  =  A  ->  A  =  C ) )
109, 4sylibrd 225 . . 3  |-  ( A  e.  B  ->  ( C  =  A  ->  [. A  /  x ]. x  =  C )
)
1110, 2syl6ibr 218 . 2  |-  ( A  e.  B  ->  ( C  =  A  ->  [. A  /  x ]. C  =  x )
)
127, 11impbid 183 1  |-  ( A  e.  B  ->  ( [. A  /  x ]. C  =  x  <->  C  =  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   [.wsbc 2991
This theorem is referenced by:  sbcoreleleq  28298  sbcoreleleqVD  28635
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-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992
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