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Theorem cbvopab 3856
 Description: Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 14-Sep-2003.)
Hypotheses
Ref Expression
cbvopab.1
cbvopab.2
cbvopab.3
cbvopab.4
cbvopab.5
Assertion
Ref Expression
cbvopab
Distinct variable group:   ,,,
Allowed substitution hints:   (,,,)   (,,,)

Proof of Theorem cbvopab
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfv 1437 . . . . 5
2 cbvopab.1 . . . . 5
31, 2nfan 1473 . . . 4
4 nfv 1437 . . . . 5
5 cbvopab.2 . . . . 5
64, 5nfan 1473 . . . 4
7 nfv 1437 . . . . 5
8 cbvopab.3 . . . . 5
97, 8nfan 1473 . . . 4
10 nfv 1437 . . . . 5
11 cbvopab.4 . . . . 5
1210, 11nfan 1473 . . . 4
13 opeq12 3579 . . . . . 6
1413eqeq2d 2067 . . . . 5
15 cbvopab.5 . . . . 5
1614, 15anbi12d 450 . . . 4
173, 6, 9, 12, 16cbvex2 1813 . . 3
1817abbii 2169 . 2
19 df-opab 3847 . 2
20 df-opab 3847 . 2
2118, 19, 203eqtr4i 2086 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 101   wb 102   wceq 1259  wnf 1365  wex 1397  cab 2042  cop 3406  copab 3845 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412  df-opab 3847 This theorem is referenced by:  cbvopabv  3857  opelopabsb  4025
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