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Theorem ineq12 3529
Description: Equality theorem for intersection of two classes. (Contributed by NM, 8-May-1994.)
Assertion
Ref Expression
ineq12  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  i^i  C
)  =  ( B  i^i  D ) )

Proof of Theorem ineq12
StepHypRef Expression
1 ineq1 3527 . 2  |-  ( A  =  B  ->  ( A  i^i  C )  =  ( B  i^i  C
) )
2 ineq2 3528 . 2  |-  ( C  =  D  ->  ( B  i^i  C )  =  ( B  i^i  D
) )
31, 2sylan9eq 2487 1  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  i^i  C
)  =  ( B  i^i  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    i^i cin 3311
This theorem is referenced by:  ineq12i  3532  ineq12d  3535  ineqan12d  3536  fnun  5543  undifixp  7090  endisj  7187  sbthlem8  7216  fiin  7419  pm54.43  7879  kmlem9  8030  indistopon  17057  epttop  17065  restbas  17214  ordtbas2  17247  txbas  17591  ptbasin  17601  trfbas2  17867  snfil  17888  fbasrn  17908  trfil2  17911  fmfnfmlem3  17980  ustuqtop2  18264  minveclem3b  19321  frrlem4  25577  diophin  26822  kelac2lem  27130
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-in 3319
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