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Theorem iunopab 4737
 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 4709 . . . . 5
21rexbii 2881 . . . 4
3 rexcom4 3053 . . . . 5
4 rexcom4 3053 . . . . . . 7
5 r19.42v 2931 . . . . . . . 8
65exbii 1726 . . . . . . 7
74, 6bitri 257 . . . . . 6
87exbii 1726 . . . . 5
93, 8bitri 257 . . . 4
102, 9bitri 257 . . 3
1110abbii 2587 . 2
12 df-iun 4271 . 2
13 df-opab 4455 . 2
1411, 12, 133eqtr4i 2503 1
 Colors of variables: wff setvar class Syntax hints:   wa 376   wceq 1452  wex 1671   wcel 1904  cab 2457  wrex 2757  cop 3965  ciun 4269  copab 4453 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-iun 4271  df-opab 4455 This theorem is referenced by:  marypha2lem2  7968
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