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Theorem eqsbc3r 2846
Description: eqsbc3 2825 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 2825 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. x  =  B  <->  A  =  B ) )
2 eqcom 2058 . . 3  |-  ( B  =  x  <->  x  =  B )
32sbcbii 2845 . 2  |-  ( [. A  /  x ]. B  =  x  <->  [. A  /  x ]. x  =  B
)
4 eqcom 2058 . 2  |-  ( B  =  A  <->  A  =  B )
51, 3, 43bitr4g 216 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  x  <->  B  =  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102    = wceq 1259    e. wcel 1409   [.wsbc 2787
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-sbc 2788
This theorem is referenced by: (None)
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