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Theorem iuneq2 3673
Description: Equality theorem for indexed union. (Contributed by NM, 22-Oct-2003.)
Assertion
Ref Expression
iuneq2  |-  ( A. x  e.  A  B  =  C  ->  U_ x  e.  A  B  =  U_ x  e.  A  C
)

Proof of Theorem iuneq2
StepHypRef Expression
1 ss2iun 3672 . . 3  |-  ( A. x  e.  A  B  C_  C  ->  U_ x  e.  A  B  C_  U_ x  e.  A  C )
2 ss2iun 3672 . . 3  |-  ( A. x  e.  A  C  C_  B  ->  U_ x  e.  A  C  C_  U_ x  e.  A  B )
31, 2anim12i 321 . 2  |-  ( ( A. x  e.  A  B  C_  C  /\  A. x  e.  A  C  C_  B )  ->  ( U_ x  e.  A  B  C_  U_ x  e.  A  C  /\  U_ x  e.  A  C  C_ 
U_ x  e.  A  B ) )
4 eqss 2960 . . . 4  |-  ( B  =  C  <->  ( B  C_  C  /\  C  C_  B ) )
54ralbii 2330 . . 3  |-  ( A. x  e.  A  B  =  C  <->  A. x  e.  A  ( B  C_  C  /\  C  C_  B ) )
6 r19.26 2441 . . 3  |-  ( A. x  e.  A  ( B  C_  C  /\  C  C_  B )  <->  ( A. x  e.  A  B  C_  C  /\  A. x  e.  A  C  C_  B
) )
75, 6bitri 173 . 2  |-  ( A. x  e.  A  B  =  C  <->  ( A. x  e.  A  B  C_  C  /\  A. x  e.  A  C  C_  B ) )
8 eqss 2960 . 2  |-  ( U_ x  e.  A  B  =  U_ x  e.  A  C 
<->  ( U_ x  e.  A  B  C_  U_ x  e.  A  C  /\  U_ x  e.  A  C  C_ 
U_ x  e.  A  B ) )
93, 7, 83imtr4i 190 1  |-  ( A. x  e.  A  B  =  C  ->  U_ x  e.  A  B  =  U_ x  e.  A  C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    = wceq 1243   A.wral 2306    C_ wss 2917   U_ciun 3657
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-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-iun 3659
This theorem is referenced by:  iuneq2i  3675  iuneq2dv  3678  dfmptg  5342
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