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Theorem xpeq1i 5019
Description: Equality inference for Cartesian product. (Contributed by NM, 21-Dec-2008.)
Hypothesis
Ref Expression
xpeq1i.1  |-  A  =  B
Assertion
Ref Expression
xpeq1i  |-  ( A  X.  C )  =  ( B  X.  C
)

Proof of Theorem xpeq1i
StepHypRef Expression
1 xpeq1i.1 . 2  |-  A  =  B
2 xpeq1 5013 . 2  |-  ( A  =  B  ->  ( A  X.  C )  =  ( B  X.  C
) )
31, 2ax-mp 5 1  |-  ( A  X.  C )  =  ( B  X.  C
)
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:  iunxpconst  5055  xpindi  5134  difxp2  5431  resdmres  5496  curry2  6875  mapsnconst  7461  mapsncnv  7462  cda1dif  8552  cdaassen  8558  infcda1  8569  geomulcvg  13644  hofcl  15382  evlsval  17959  matvsca2  18697  ovoliunnul  21653  vitalilem5  21756  lgam1  28246  mendvscafval  30744  xpprsng  31985
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