ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbceq1g Unicode version

Theorem sbceq1g 2898
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 2894 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  C  <->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) )
2 csbconstg 2892 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ C  =  C )
32eqeq2d 2067 . 2  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ B  =  [_ A  /  x ]_ C  <->  [_ A  /  x ]_ B  =  C ) )
41, 3bitrd 181 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  C  <->  [_ A  /  x ]_ B  =  C )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102    = wceq 1259    e. wcel 1409   [.wsbc 2787   [_csb 2880
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  df-csb 2881
This theorem is referenced by:  f1od2  5884
  Copyright terms: Public domain W3C validator