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Theorem iuneq2 3741
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 3740 . . 3  |-  ( A. x  e.  A  B  C_  C  ->  U_ x  e.  A  B  C_  U_ x  e.  A  C )
2 ss2iun 3740 . . 3  |-  ( A. x  e.  A  C  C_  B  ->  U_ x  e.  A  C  C_  U_ x  e.  A  B )
31, 2anim12i 331 . 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 3038 . . . 4  |-  ( B  =  C  <->  ( B  C_  C  /\  C  C_  B ) )
54ralbii 2384 . . 3  |-  ( A. x  e.  A  B  =  C  <->  A. x  e.  A  ( B  C_  C  /\  C  C_  B ) )
6 r19.26 2497 . . 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 182 . 2  |-  ( A. x  e.  A  B  =  C  <->  ( A. x  e.  A  B  C_  C  /\  A. x  e.  A  C  C_  B ) )
8 eqss 3038 . 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 199 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 102    = wceq 1289   A.wral 2359    C_ wss 2997   U_ciun 3725
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-in 3003  df-ss 3010  df-iun 3727
This theorem is referenced by:  iuneq2i  3743  iuneq2dv  3746  dfmptg  5460
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