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Theorem sbss 3563
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 2791 . 2  |-  y  e. 
_V
2 sbequ 2000 . 2  |-  ( z  =  y  ->  ( [ z  /  x ] x  C_  A  <->  [ y  /  x ] x  C_  A ) )
3 sseq1 3199 . 2  |-  ( z  =  y  ->  (
z  C_  A  <->  y  C_  A ) )
4 nfv 1605 . . 3  |-  F/ x  z  C_  A
5 sseq1 3199 . . 3  |-  ( x  =  z  ->  (
x  C_  A  <->  z  C_  A ) )
64, 5sbie 1978 . 2  |-  ( [ z  /  x ]
x  C_  A  <->  z  C_  A )
71, 2, 3, 6vtoclb 2841 1  |-  ( [ y  /  x ]
x  C_  A  <->  y  C_  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   [wsb 1629    C_ wss 3152
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-v 2790  df-in 3159  df-ss 3166
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