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Theorem sbcel12g 3151
 Description: Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
sbcel12g

Proof of Theorem sbcel12g
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3049 . . 3
2 dfsbcq2 3049 . . . . 5
32abbidv 2467 . . . 4
4 dfsbcq2 3049 . . . . 5
54abbidv 2467 . . . 4
63, 5eleq12d 2421 . . 3
7 nfs1v 2106 . . . . . 6
87nfab 2493 . . . . 5
9 nfs1v 2106 . . . . . 6
109nfab 2493 . . . . 5
118, 10nfel 2497 . . . 4
12 sbab 2475 . . . . 5
13 sbab 2475 . . . . 5
1412, 13eleq12d 2421 . . . 4
1511, 14sbie 2038 . . 3
161, 6, 15vtoclbg 2915 . 2
17 df-csb 3137 . . 3
18 df-csb 3137 . . 3
1917, 18eleq12i 2418 . 2
2016, 19syl6bbr 254 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176   wceq 1642  wsb 1648   wcel 1710  cab 2339  wsbc 3046  csb 3136 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-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 This theorem is referenced by:  sbcnel12g  3153  sbcel1g  3155  sbcel2g  3157  sbccsb2g  3165
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