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Theorem eqsbc3r 2899
Description: eqsbc3 2878 with setvar variable on right side of equals sign. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by JJ, 7-Jul-2021.)
Assertion
Ref Expression
eqsbc3r  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  x  <->  B  =  A ) )
Distinct variable group:    x, B
Allowed substitution hints:    A( x)    V( x)

Proof of Theorem eqsbc3r
StepHypRef Expression
1 eqsbc3 2878 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. x  =  B  <->  A  =  B ) )
2 eqcom 2090 . . 3  |-  ( B  =  x  <->  x  =  B )
32sbcbii 2898 . 2  |-  ( [. A  /  x ]. B  =  x  <->  [. A  /  x ]. x  =  B
)
4 eqcom 2090 . 2  |-  ( B  =  A  <->  A  =  B )
51, 3, 43bitr4g 221 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  x  <->  B  =  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1289    e. wcel 1438   [.wsbc 2840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2841
This theorem is referenced by: (None)
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