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Theorem eqsbc3r 3291
Description: eqsbc3 3274 with setvar variable on right side of equals sign. This proof was automatically generated from the virtual deduction proof eqsbc3rVD 37232 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 2458 . . . . . 6  |-  ( B  =  x  <->  x  =  B )
21sbcbii 3290 . . . . 5  |-  ( [. A  /  x ]. B  =  x  <->  [. A  /  x ]. x  =  B
)
32biimpi 199 . . . 4  |-  ( [. A  /  x ]. B  =  x  ->  [. A  /  x ]. x  =  B )
4 eqsbc3 3274 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. x  =  B  <->  A  =  B ) )
53, 4syl5ib 227 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  x  ->  A  =  B ) )
6 eqcom 2458 . . 3  |-  ( A  =  B  <->  B  =  A )
75, 6syl6ib 234 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  x  ->  B  =  A ) )
8 idd 25 . . . . 5  |-  ( A  e.  V  ->  ( B  =  A  ->  B  =  A ) )
98, 6syl6ibr 235 . . . 4  |-  ( A  e.  V  ->  ( B  =  A  ->  A  =  B ) )
109, 4sylibrd 242 . . 3  |-  ( A  e.  V  ->  ( B  =  A  ->  [. A  /  x ]. x  =  B )
)
1110, 2syl6ibr 235 . 2  |-  ( A  e.  V  ->  ( B  =  A  ->  [. A  /  x ]. B  =  x )
)
127, 11impbid 195 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  x  <->  B  =  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    = wceq 1447    e. wcel 1890   [.wsbc 3234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-10 1918  ax-11 1923  ax-12 1936  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1450  df-ex 1667  df-nf 1671  df-sb 1801  df-clab 2438  df-cleq 2444  df-clel 2447  df-v 3014  df-sbc 3235
This theorem is referenced by:  sbcoreleleq  36896  sbcoreleleqVD  37252
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