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Theorem cbvopab2 4009
 Description: Change second bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 22-Aug-2013.)
Hypotheses
Ref Expression
cbvopab2.1
cbvopab2.2
cbvopab2.3
Assertion
Ref Expression
cbvopab2
Distinct variable group:   ,,
Allowed substitution hints:   (,,)   (,,)

Proof of Theorem cbvopab2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfv 1509 . . . . . 6
2 cbvopab2.1 . . . . . 6
31, 2nfan 1545 . . . . 5
4 nfv 1509 . . . . . 6
5 cbvopab2.2 . . . . . 6
64, 5nfan 1545 . . . . 5
7 opeq2 3713 . . . . . . 7
87eqeq2d 2152 . . . . . 6
9 cbvopab2.3 . . . . . 6
108, 9anbi12d 465 . . . . 5
113, 6, 10cbvex 1730 . . . 4
1211exbii 1585 . . 3
1312abbii 2256 . 2
14 df-opab 3997 . 2
15 df-opab 3997 . 2
1613, 14, 153eqtr4i 2171 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wb 104   wceq 1332  wnf 1437  wex 1469  cab 2126  cop 3534  copab 3995 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3079  df-sn 3537  df-pr 3538  df-op 3540  df-opab 3997 This theorem is referenced by:  cbvoprab3  5854
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