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Theorem ineq12 3378
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 3376 . 2  |-  ( A  =  B  ->  ( A  i^i  C )  =  ( B  i^i  C
) )
2 ineq2 3377 . 2  |-  ( C  =  D  ->  ( B  i^i  C )  =  ( B  i^i  D
) )
31, 2sylan9eq 2348 1  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  i^i  C
)  =  ( B  i^i  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    i^i cin 3164
This theorem is referenced by:  ineq12i  3381  ineq12d  3384  ineqan12d  3385  fnun  5366  undifixp  6868  endisj  6965  sbthlem8  6994  fiin  7191  pm54.43  7649  kmlem9  7800  indistopon  16754  epttop  16762  restbas  16905  ordtbas2  16937  txbas  17278  ptbasin  17288  trfbas2  17554  snfil  17575  fbasrn  17595  trfil2  17598  fmfnfmlem3  17667  minveclem3b  18808  frrlem4  24355  diophin  26955  kelac2lem  27265
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-in 3172
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