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Theorem sbss 3913
 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
Distinct variable group:   ,
Allowed substitution hint:   ()

Proof of Theorem sbss
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 vex 3090 . 2
2 sbequ 2171 . 2
3 sseq1 3491 . 2
4 nfv 1754 . . 3
5 sseq1 3491 . . 3
64, 5sbie 2203 . 2
71, 2, 3, 6vtoclb 3142 1
 Colors of variables: wff setvar class Syntax hints:   wb 187  wsb 1789   wss 3442 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-v 3089  df-in 3449  df-ss 3456 This theorem is referenced by: (None)
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