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Theorem sbcssg 3940
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 sbcal 3379 . . 3
2 sbcimg 3369 . . . . 5
3 sbcel2 3831 . . . . . 6
4 sbcel2 3831 . . . . . 6
53, 4imbi12i 326 . . . . 5
62, 5syl6bb 261 . . . 4
76albidv 1713 . . 3
81, 7syl5bb 257 . 2
9 dfss2 3492 . . 3
109sbcbii 3387 . 2
11 dfss2 3492 . 2
128, 10, 113bitr4g 288 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  A.wal 1393  e.wcel 1818  [.wsbc 3327  [_csb 3434  C_wss 3475
This theorem is referenced by:  sbcrel  5094  sbcfg  5734  iuninc  27428
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-fal 1401  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-in 3482  df-ss 3489  df-nul 3785
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