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Theorem bj-sbeqALT 33548
Description: Substitution in an equality (use the more genereal version bj-sbeq 33549 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 3451 . . 3  |-  F/_ x [_ y  /  x ]_ A
2 nfcsb1v 3451 . . 3  |-  F/_ x [_ y  /  x ]_ B
31, 2nfeq 2640 . 2  |-  F/ x [_ y  /  x ]_ A  =  [_ y  /  x ]_ B
4 csbeq1a 3444 . . 3  |-  ( x  =  y  ->  A  =  [_ y  /  x ]_ A )
5 csbeq1a 3444 . . 3  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
64, 5eqeq12d 2489 . 2  |-  ( x  =  y  ->  ( A  =  B  <->  [_ y  /  x ]_ A  =  [_ y  /  x ]_ B
) )
73, 6sbie 2123 1  |-  ( [ y  /  x ] A  =  B  <->  [_ y  /  x ]_ A  =  [_ y  /  x ]_ B
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1379   [wsb 1711   [_csb 3435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-sbc 3332  df-csb 3436
This theorem is referenced by: (None)
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