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Theorem sbss 3730
 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 2952 . 2
2 sbequ 2140 . 2
3 sseq1 3362 . 2
4 nfv 1629 . . 3
5 sseq1 3362 . . 3
64, 5sbie 2124 . 2
71, 2, 3, 6vtoclb 3002 1
 Colors of variables: wff set class Syntax hints:   wb 177  wsb 1658   wss 3313 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417 This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-v 2951  df-in 3320  df-ss 3327
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