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Theorem sbss 3939
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:   ,

Proof of Theorem sbss
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 vex 3112 . 2
2 sbequ 2117 . 2
3 sseq1 3524 . 2
4 nfv 1707 . . 3
5 sseq1 3524 . . 3
64, 5sbie 2149 . 2
71, 2, 3, 6vtoclb 3164 1
Colors of variables: wff setvar class
Syntax hints:  <->wb 184  [wsb 1739  C_wss 3475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-v 3111  df-in 3482  df-ss 3489
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