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Theorem iuneq2 4328
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 4327 . . 3  |-  ( A. x  e.  A  B  C_  C  ->  U_ x  e.  A  B  C_  U_ x  e.  A  C )
2 ss2iun 4327 . . 3  |-  ( A. x  e.  A  C  C_  B  ->  U_ x  e.  A  C  C_  U_ x  e.  A  B )
31, 2anim12i 566 . 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 3501 . . . 4  |-  ( B  =  C  <->  ( B  C_  C  /\  C  C_  B ) )
54ralbii 2872 . . 3  |-  ( A. x  e.  A  B  =  C  <->  A. x  e.  A  ( B  C_  C  /\  C  C_  B ) )
6 r19.26 2968 . . 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 249 . 2  |-  ( A. x  e.  A  B  =  C  <->  ( A. x  e.  A  B  C_  C  /\  A. x  e.  A  C  C_  B ) )
8 eqss 3501 . 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 266 1  |-  ( A. x  e.  A  B  =  C  ->  U_ x  e.  A  B  =  U_ x  e.  A  C
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1381   A.wral 2791    C_ wss 3458   U_ciun 4311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ral 2796  df-rex 2797  df-v 3095  df-in 3465  df-ss 3472  df-iun 4313
This theorem is referenced by:  iuneq2i  4330  iuneq2dv  4333  oa0r  7186  om0r  7187  om1r  7190  oe1m  7192  oaass  7208  oarec  7209  omass  7227  oeoalem  7243  oeoelem  7245  cardiun  8361  kmlem11  8538  iuncld  19412  comppfsc  19899  istotbnd3  30235  sstotbnd  30239  heibor  30285  iuneq12f  30540
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