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Theorem sbcss 3660
 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
sbcss

Proof of Theorem sbcss
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 sbcalg 3094 . . 3
2 sbcimg 3087 . . . . 5
3 sbcel2g 3157 . . . . . 6
4 sbcel2g 3157 . . . . . 6
53, 4imbi12d 311 . . . . 5
62, 5bitrd 244 . . . 4
76albidv 1625 . . 3
81, 7bitrd 244 . 2
9 dfss2 3262 . . 3
109sbcbii 3101 . 2
11 dfss2 3262 . 2
128, 10, 113bitr4g 279 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176  wal 1540   wcel 1710  wsbc 3046  csb 3136   wss 3257 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-sbc 3047  df-csb 3137  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259 This theorem is referenced by: (None)
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