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Theorem xpeq12 4854
Description: Equality theorem for Cartesian product. (Contributed by FL, 31-Aug-2009.)
Assertion
Ref Expression
xpeq12  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  X.  C
)  =  ( B  X.  D ) )

Proof of Theorem xpeq12
StepHypRef Expression
1 xpeq1 4849 . 2  |-  ( A  =  B  ->  ( A  X.  C )  =  ( B  X.  C
) )
2 xpeq2 4850 . 2  |-  ( C  =  D  ->  ( B  X.  C )  =  ( B  X.  D
) )
31, 2sylan9eq 2490 1  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  X.  C
)  =  ( B  X.  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    X. cxp 4833
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-opab 4346  df-xp 4841
This theorem is referenced by:  xpeq12i  4857  xpeq12d  4860  xpid11  5056  xp11  5268  infxpenlem  8172  fpwwe2lem5  8793  pwfseqlem4a  8820  pwfseqlem4  8821  pwfseqlem5  8822  pwfseq  8823  pwsval  14416  mamufval  18258  mvmulfval  18328  txtopon  19139  txbasval  19154  txindislem  19181  ismet  19873  isxmet  19874  ismgm  23758  opidon2  23762  shsval  24666  prdsbnd2  28647  ttac  29338  sblpnf  29549
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