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Theorem bj-sbeqALT 31031
Description: Substitution in an equality (use the more genereal version bj-sbeq 31032 instead, without disjoint variable condition). (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.) (Proof modification 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 3389 . . 3  |-  F/_ x [_ y  /  x ]_ A
2 nfcsb1v 3389 . . 3  |-  F/_ x [_ y  /  x ]_ B
31, 2nfeq 2575 . 2  |-  F/ x [_ y  /  x ]_ A  =  [_ y  /  x ]_ B
4 csbeq1a 3382 . . 3  |-  ( x  =  y  ->  A  =  [_ y  /  x ]_ A )
5 csbeq1a 3382 . . 3  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
64, 5eqeq12d 2424 . 2  |-  ( x  =  y  ->  ( A  =  B  <->  [_ y  /  x ]_ A  =  [_ y  /  x ]_ B
) )
73, 6sbie 2173 1  |-  ( [ y  /  x ] A  =  B  <->  [_ y  /  x ]_ A  =  [_ y  /  x ]_ B
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1405   [wsb 1763   [_csb 3373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-sbc 3278  df-csb 3374
This theorem is referenced by: (None)
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