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Theorem sbcss12g 23157
Description: Set substitution into the both argument of a subset relation. (Contributed by Thierry Arnoux, 25-Jan-2017.)
Assertion
Ref Expression
sbcss12g  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  C_  C  <->  [_ A  /  x ]_ B  C_  [_ A  /  x ]_ C ) )
Distinct variable group:    x, A
Allowed substitution hints:    B( x)    C( x)    V( x)

Proof of Theorem sbcss12g
StepHypRef Expression
1 nfcsb1v 3126 . . 3  |-  F/_ x [_ A  /  x ]_ B
2 nfcsb1v 3126 . . 3  |-  F/_ x [_ A  /  x ]_ C
31, 2nfss 3186 . 2  |-  F/ x [_ A  /  x ]_ B  C_  [_ A  /  x ]_ C
4 csbeq1a 3102 . . 3  |-  ( x  =  A  ->  B  =  [_ A  /  x ]_ B )
5 csbeq1a 3102 . . 3  |-  ( x  =  A  ->  C  =  [_ A  /  x ]_ C )
64, 5sseq12d 3220 . 2  |-  ( x  =  A  ->  ( B  C_  C  <->  [_ A  /  x ]_ B  C_  [_ A  /  x ]_ C ) )
73, 6sbciegf 3035 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  C_  C  <->  [_ A  /  x ]_ B  C_  [_ A  /  x ]_ C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696   [.wsbc 3004   [_csb 3094    C_ wss 3165
This theorem is referenced by:  iuninc  23174
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-v 2803  df-sbc 3005  df-csb 3095  df-in 3172  df-ss 3179
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