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Theorem sbss 3522
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 2733 . 2  |-  y  e. 
_V
2 sbequ 1833 . 2  |-  ( z  =  y  ->  ( [ z  /  x ] x  C_  A  <->  [ y  /  x ] x  C_  A ) )
3 sseq1 3170 . 2  |-  ( z  =  y  ->  (
z  C_  A  <->  y  C_  A ) )
4 nfv 1521 . . 3  |-  F/ x  z  C_  A
5 sseq1 3170 . . 3  |-  ( x  =  z  ->  (
x  C_  A  <->  z  C_  A ) )
64, 5sbie 1784 . 2  |-  ( [ z  /  x ]
x  C_  A  <->  z  C_  A )
71, 2, 3, 6vtoclb 2787 1  |-  ( [ y  /  x ]
x  C_  A  <->  y  C_  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   [wsb 1755    C_ wss 3121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732  df-in 3127  df-ss 3134
This theorem is referenced by: (None)
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