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Theorem xpeq2i 5020
Description: Equality inference for Cartesian product. (Contributed by NM, 21-Dec-2008.)
Hypothesis
Ref Expression
xpeq1i.1  |-  A  =  B
Assertion
Ref Expression
xpeq2i  |-  ( C  X.  A )  =  ( C  X.  B
)

Proof of Theorem xpeq2i
StepHypRef Expression
1 xpeq1i.1 . 2  |-  A  =  B
2 xpeq2 5014 . 2  |-  ( A  =  B  ->  ( C  X.  A )  =  ( C  X.  B
) )
31, 2ax-mp 5 1  |-  ( C  X.  A )  =  ( C  X.  B
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    X. cxp 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-opab 4506  df-xp 5005
This theorem is referenced by:  xpindir  5137  difxp1  5432  xpima  5449  xpexgALT  6777  curry1  6875  fparlem3  6885  fparlem4  6886  xp1en  7603  xp2cda  8560  xpcdaen  8563  pwcda1  8574  pwcdandom  9045  yonedalem3b  15406  yonedalem3  15407  pws1  17066  pwsmgp  17068  xkoinjcn  19951  imasdsf1olem  20639  zrdivrng  25138  df0op2  26375  ho01i  26451  nmop0h  26614  mbfmcst  27898  0rrv  28058  cvmlift2lem12  28427
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