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Theorem ineq12 3473
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 3471 . 2  |-  ( A  =  B  ->  ( A  i^i  C )  =  ( B  i^i  C
) )
2 ineq2 3472 . 2  |-  ( C  =  D  ->  ( B  i^i  C )  =  ( B  i^i  D
) )
31, 2sylan9eq 2432 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 1649    i^i cin 3255
This theorem is referenced by:  ineq12i  3476  ineq12d  3479  ineqan12d  3480  fnun  5484  undifixp  7027  endisj  7124  sbthlem8  7153  fiin  7355  pm54.43  7813  kmlem9  7964  indistopon  16981  epttop  16989  restbas  17137  ordtbas2  17170  txbas  17513  ptbasin  17523  trfbas2  17789  snfil  17810  fbasrn  17830  trfil2  17833  fmfnfmlem3  17902  ustuqtop2  18186  minveclem3b  19189  frrlem4  25301  diophin  26515  kelac2lem  26824
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-v 2894  df-in 3263
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