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Theorem opelopabsbALT 4042
 Description: The law of concretion in terms of substitutions. Less general than opelopabsb 4043, but having a much shorter proof. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
opelopabsbALT
Distinct variable groups:   ,,   ,,
Allowed substitution hints:   (,,,)

Proof of Theorem opelopabsbALT
StepHypRef Expression
1 excom 1595 . . 3
2 vex 2613 . . . . . . 7
3 vex 2613 . . . . . . 7
42, 3opth 4020 . . . . . 6
5 equcom 1635 . . . . . . 7
6 equcom 1635 . . . . . . 7
75, 6anbi12ci 449 . . . . . 6
84, 7bitri 182 . . . . 5
98anbi1i 446 . . . 4
1092exbii 1538 . . 3
111, 10bitri 182 . 2
12 elopab 4041 . 2
13 2sb5 1902 . 2
1411, 12, 133bitr4i 210 1
 Colors of variables: wff set class Syntax hints:   wa 102   wb 103   wceq 1285  wex 1422   wcel 1434  wsb 1687  cop 3419  copab 3858 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-opab 3860 This theorem is referenced by:  inopab  4516  cnvopab  4776
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