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Theorem eqsbc3r 3243
Description: eqsbc3 3221 with setvar variable on right side of equals sign. This proof was automatically generated from the virtual deduction proof eqsbc3rVD 31463 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.)
Assertion
Ref Expression
eqsbc3r  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  x  <->  B  =  A ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    V( x)

Proof of Theorem eqsbc3r
StepHypRef Expression
1 eqcom 2440 . . . . . 6  |-  ( B  =  x  <->  x  =  B )
21sbcbii 3241 . . . . 5  |-  ( [. A  /  x ]. B  =  x  <->  [. A  /  x ]. x  =  B
)
32biimpi 194 . . . 4  |-  ( [. A  /  x ]. B  =  x  ->  [. A  /  x ]. x  =  B )
4 eqsbc3 3221 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. x  =  B  <->  A  =  B ) )
53, 4syl5ib 219 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  x  ->  A  =  B ) )
6 eqcom 2440 . . 3  |-  ( A  =  B  <->  B  =  A )
75, 6syl6ib 226 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  x  ->  B  =  A ) )
8 idd 24 . . . . 5  |-  ( A  e.  V  ->  ( B  =  A  ->  B  =  A ) )
98, 6syl6ibr 227 . . . 4  |-  ( A  e.  V  ->  ( B  =  A  ->  A  =  B ) )
109, 4sylibrd 234 . . 3  |-  ( A  e.  V  ->  ( B  =  A  ->  [. A  /  x ]. x  =  B )
)
1110, 2syl6ibr 227 . 2  |-  ( A  e.  V  ->  ( B  =  A  ->  [. A  /  x ]. B  =  x )
)
127, 11impbid 191 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  x  <->  B  =  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1369    e. wcel 1756   [.wsbc 3181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-v 2969  df-sbc 3182
This theorem is referenced by:  sbcoreleleq  31128  sbcoreleleqVD  31482
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