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Theorem sbss 3366
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 2613 . 2  |-  y  e. 
_V
2 sbequ 1763 . 2  |-  ( z  =  y  ->  ( [ z  /  x ] x  C_  A  <->  [ y  /  x ] x  C_  A ) )
3 sseq1 3029 . 2  |-  ( z  =  y  ->  (
z  C_  A  <->  y  C_  A ) )
4 nfv 1462 . . 3  |-  F/ x  z  C_  A
5 sseq1 3029 . . 3  |-  ( x  =  z  ->  (
x  C_  A  <->  z  C_  A ) )
64, 5sbie 1716 . 2  |-  ( [ z  /  x ]
x  C_  A  <->  z  C_  A )
71, 2, 3, 6vtoclb 2665 1  |-  ( [ y  /  x ]
x  C_  A  <->  y  C_  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 103   [wsb 1687    C_ wss 2982
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-v 2612  df-in 2988  df-ss 2995
This theorem is referenced by: (None)
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