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Theorem sbabel 2203
 Description: Theorem to move a substitution in and out of a class abstraction. (Contributed by NM, 27-Sep-2003.) (Revised by Mario Carneiro, 7-Oct-2016.)
Hypothesis
Ref Expression
sbabel.1
Assertion
Ref Expression
sbabel
Distinct variable groups:   ,   ,
Allowed substitution hints:   (,,)   (,,)

Proof of Theorem sbabel
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 sbex 1880 . . 3
2 sban 1829 . . . . 5
3 nfv 1421 . . . . . . . . . 10
43sbf 1660 . . . . . . . . 9
54sbrbis 1835 . . . . . . . 8
65sbalv 1881 . . . . . . 7
7 abeq2 2146 . . . . . . . 8
87sbbii 1648 . . . . . . 7
9 abeq2 2146 . . . . . . 7
106, 8, 93bitr4i 201 . . . . . 6
11 sbabel.1 . . . . . . . 8
1211nfcri 2172 . . . . . . 7
1312sbf 1660 . . . . . 6
1410, 13anbi12i 433 . . . . 5
152, 14bitri 173 . . . 4
1615exbii 1496 . . 3
171, 16bitri 173 . 2
18 df-clel 2036 . . 3
1918sbbii 1648 . 2
20 df-clel 2036 . 2
2117, 19, 203bitr4i 201 1
 Colors of variables: wff set class Syntax hints:   wa 97   wb 98  wal 1241   wceq 1243  wex 1381   wcel 1393  wsb 1645  cab 2026  wnfc 2165 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167 This theorem is referenced by: (None)
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