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Theorem sbss 3937
Description: Set substitution into the first argument of a subset relation. (Contributed by Rodolfo Medina, 7-Jul-2010.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
sbss  |-  ( [ y  /  x ]
x  C_  A  <->  y  C_  A )
Distinct variable group:    x, A
Allowed substitution hint:    A( y)

Proof of Theorem sbss
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 vex 3116 . 2  |-  y  e. 
_V
2 sbequ 2090 . 2  |-  ( z  =  y  ->  ( [ z  /  x ] x  C_  A  <->  [ y  /  x ] x  C_  A ) )
3 sseq1 3525 . 2  |-  ( z  =  y  ->  (
z  C_  A  <->  y  C_  A ) )
4 nfv 1683 . . 3  |-  F/ x  z  C_  A
5 sseq1 3525 . . 3  |-  ( x  =  z  ->  (
x  C_  A  <->  z  C_  A ) )
64, 5sbie 2123 . 2  |-  ( [ z  /  x ]
x  C_  A  <->  z  C_  A )
71, 2, 3, 6vtoclb 3168 1  |-  ( [ y  /  x ]
x  C_  A  <->  y  C_  A )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   [wsb 1711    C_ wss 3476
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-v 3115  df-in 3483  df-ss 3490
This theorem is referenced by: (None)
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