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Theorem sbcssg 3472
 Description: Distribute proper substitution through a subclass relation. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Alexander van der Vekens, 23-Jul-2017.)
Assertion
Ref Expression
sbcssg

Proof of Theorem sbcssg
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 sbcalg 2961 . . 3
2 sbcimg 2950 . . . . 5
3 sbcel2g 3023 . . . . . 6
4 sbcel2g 3023 . . . . . 6
53, 4imbi12d 233 . . . . 5
62, 5bitrd 187 . . . 4
76albidv 1796 . . 3
81, 7bitrd 187 . 2
9 dfss2 3086 . . 3
109sbcbii 2968 . 2
11 dfss2 3086 . 2
128, 10, 113bitr4g 222 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 104  wal 1329   wcel 1480  wsbc 2909  csb 3003   wss 3071 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-sbc 2910  df-csb 3004  df-in 3077  df-ss 3084 This theorem is referenced by:  sbcrel  4625  sbcfg  5271
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