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Theorem iunopab 4018
 Description: Move indexed union inside an ordered-pair abstraction. (Contributed by Stefan O'Rear, 20-Feb-2015.)
Assertion
Ref Expression
iunopab
Distinct variable groups:   ,   ,   ,   ,
Allowed substitution hints:   (,,)   ()

Proof of Theorem iunopab
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elopab 3995 . . . . 5
21rexbii 2331 . . . 4
3 rexcom4 2577 . . . . 5
4 rexcom4 2577 . . . . . . 7
5 r19.42v 2467 . . . . . . . 8
65exbii 1496 . . . . . . 7
74, 6bitri 173 . . . . . 6
87exbii 1496 . . . . 5
93, 8bitri 173 . . . 4
102, 9bitri 173 . . 3
1110abbii 2153 . 2
12 df-iun 3659 . 2
13 df-opab 3819 . 2
1411, 12, 133eqtr4i 2070 1
 Colors of variables: wff set class Syntax hints:   wa 97   wceq 1243  wex 1381   wcel 1393  cab 2026  wrex 2307  cop 3378  ciun 3657  copab 3817 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-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-iun 3659  df-opab 3819 This theorem is referenced by: (None)
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