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Theorem sbcss 3730
 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 3201 . . 3
2 sbcimg 3194 . . . . 5
3 sbcel2g 3264 . . . . . 6
4 sbcel2g 3264 . . . . . 6
53, 4imbi12d 312 . . . . 5
62, 5bitrd 245 . . . 4
76albidv 1635 . . 3
81, 7bitrd 245 . 2
9 dfss2 3329 . . 3
109sbcbii 3208 . 2
11 dfss2 3329 . 2
128, 10, 113bitr4g 280 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177  wal 1549   wcel 1725  wsbc 3153  csb 3243   wss 3312 This theorem is referenced by:  iuninc  24003  sbcrel  27948 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 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-sbc 3154  df-csb 3244  df-in 3319  df-ss 3326
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