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Theorem iunopab 4487
 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 4463 . . . . 5
21rexbii 2731 . . . 4
3 rexcom4 2976 . . . . 5
4 rexcom4 2976 . . . . . . 7
5 r19.42v 2863 . . . . . . . 8
65exbii 1593 . . . . . . 7
74, 6bitri 242 . . . . . 6
87exbii 1593 . . . . 5
93, 8bitri 242 . . . 4
102, 9bitri 242 . . 3
1110abbii 2549 . 2
12 df-iun 4096 . 2
13 df-opab 4268 . 2
1411, 12, 133eqtr4i 2467 1
 Colors of variables: wff set class Syntax hints:   wa 360  wex 1551   wceq 1653   wcel 1726  cab 2423  wrex 2707  cop 3818  ciun 4094  copab 4266 This theorem is referenced by:  marypha2lem2  7442 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-rex 2712  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-iun 4096  df-opab 4268
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