Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-sbeqALT Structured version   Unicode version

Theorem bj-sbeqALT 32402
Description: Substitution in an equality (use the more genereal version bj-sbeq 32403 instead, without disjoint variable condition). (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.)
Assertion
Ref Expression
bj-sbeqALT  |-  ( [ y  /  x ] A  =  B  <->  [_ y  /  x ]_ A  =  [_ y  /  x ]_ B
)
Distinct variable group:    x, y
Allowed substitution hints:    A( x, y)    B( x, y)

Proof of Theorem bj-sbeqALT
StepHypRef Expression
1 nfcsb1v 3304 . . 3  |-  F/_ x [_ y  /  x ]_ A
2 nfcsb1v 3304 . . 3  |-  F/_ x [_ y  /  x ]_ B
31, 2nfeq 2586 . 2  |-  F/ x [_ y  /  x ]_ A  =  [_ y  /  x ]_ B
4 csbeq1a 3297 . . 3  |-  ( x  =  y  ->  A  =  [_ y  /  x ]_ A )
5 csbeq1a 3297 . . 3  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
64, 5eqeq12d 2457 . 2  |-  ( x  =  y  ->  ( A  =  B  <->  [_ y  /  x ]_ A  =  [_ y  /  x ]_ B
) )
73, 6sbie 2100 1  |-  ( [ y  /  x ] A  =  B  <->  [_ y  /  x ]_ A  =  [_ y  /  x ]_ B
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1369   [wsb 1700   [_csb 3288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-sbc 3187  df-csb 3289
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator