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Theorem xpeq2i 5106
Description: Equality inference for Cartesian product. (Contributed by NM, 21-Dec-2008.)
Hypothesis
Ref Expression
xpeq1i.1 𝐴 = 𝐵
Assertion
Ref Expression
xpeq2i (𝐶 × 𝐴) = (𝐶 × 𝐵)

Proof of Theorem xpeq2i
StepHypRef Expression
1 xpeq1i.1 . 2 𝐴 = 𝐵
2 xpeq2 5099 . 2 (𝐴 = 𝐵 → (𝐶 × 𝐴) = (𝐶 × 𝐵))
31, 2ax-mp 5 1 (𝐶 × 𝐴) = (𝐶 × 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480   × cxp 5082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-opab 4684  df-xp 5090
This theorem is referenced by:  xpindir  5226  xpssres  5403  difxp1  5528  xpima  5545  xpexgALT  7121  curry1  7229  fparlem3  7239  fparlem4  7240  xp1en  8006  xp2cda  8962  xpcdaen  8965  pwcda1  8976  pwcdandom  9449  yonedalem3b  16859  yonedalem3  16860  pws1  18556  pwsmgp  18558  xkoinjcn  21430  imasdsf1olem  22118  df0op2  28499  ho01i  28575  nmop0h  28738  mbfmcst  30144  0rrv  30336  cvmlift2lem12  31057  zrdivrng  33423
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