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Theorem sbss 3576
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 2804 . 2  |-  y  e. 
_V
2 sbequ 2013 . 2  |-  ( z  =  y  ->  ( [ z  /  x ] x  C_  A  <->  [ y  /  x ] x  C_  A ) )
3 sseq1 3212 . 2  |-  ( z  =  y  ->  (
z  C_  A  <->  y  C_  A ) )
4 nfv 1609 . . 3  |-  F/ x  z  C_  A
5 sseq1 3212 . . 3  |-  ( x  =  z  ->  (
x  C_  A  <->  z  C_  A ) )
64, 5sbie 1991 . 2  |-  ( [ z  /  x ]
x  C_  A  <->  z  C_  A )
71, 2, 3, 6vtoclb 2854 1  |-  ( [ y  /  x ]
x  C_  A  <->  y  C_  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   [wsb 1638    C_ wss 3165
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-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-v 2803  df-in 3172  df-ss 3179
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