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Theorem eqsbc2 3023
Description: Substitution for the right-hand side in an equality. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by JJ, 7-Jul-2021.)
Assertion
Ref Expression
eqsbc2  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  x  <->  B  =  A ) )
Distinct variable group:    x, B
Allowed substitution hints:    A( x)    V( x)

Proof of Theorem eqsbc2
StepHypRef Expression
1 eqsbc1 3002 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. x  =  B  <->  A  =  B ) )
2 eqcom 2179 . . 3  |-  ( B  =  x  <->  x  =  B )
32sbcbii 3022 . 2  |-  ( [. A  /  x ]. B  =  x  <->  [. A  /  x ]. x  =  B
)
4 eqcom 2179 . 2  |-  ( B  =  A  <->  A  =  B )
51, 3, 43bitr4g 223 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  x  <->  B  =  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1353    e. wcel 2148   [.wsbc 2962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-sbc 2963
This theorem is referenced by: (None)
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