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Theorem sbceq1g 3022
Description: Move proper substitution to first argument of an equality. (Contributed by NM, 30-Nov-2005.)
Assertion
Ref Expression
sbceq1g  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  C  <->  [_ A  /  x ]_ B  =  C )
)
Distinct variable group:    x, C
Allowed substitution hints:    A( x)    B( x)    V( x)

Proof of Theorem sbceq1g
StepHypRef Expression
1 sbceqg 3018 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  C  <->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) )
2 csbconstg 3016 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ C  =  C )
32eqeq2d 2151 . 2  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ B  =  [_ A  /  x ]_ C  <->  [_ A  /  x ]_ B  =  C ) )
41, 3bitrd 187 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  C  <->  [_ A  /  x ]_ B  =  C )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1331    e. wcel 1480   [.wsbc 2909   [_csb 3003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-sbc 2910  df-csb 3004
This theorem is referenced by:  f1od2  6132
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