Theorem xpeq1i 5710

Description: Equality inference for Cartesian product. (Contributed by NM, 21-Dec-2008.)

Hypothesis

Ref	Expression
xpeq1i.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
xpeq1i	⊢ (𝐴 × 𝐶) = (𝐵 × 𝐶)

Proof of Theorem xpeq1i

Step	Hyp	Ref	Expression
1		xpeq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		xpeq1 5698	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐶)

Syntax hints:   = wceq 1539   × cxp 5682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-opab 5205  df-xp 5690

This theorem is referenced by:  iunxpconst  5757  xpindi  5843  difxp2  6185  resdmres  6251  xpprsng  7159  curry2  8133  mapsnconst  8933  mapsncnv  8934  xp2dju  10218  pwdju1  10232  pwdjundom  10708  geomulcvg  15913  hofcl  18305  evlsval  22111  matvsca2  22435  ehl0  25452  ovoliunnul  25543  vitalilem5  25648  lgam1  27108  iunxpssiun1  32582  finxp2o  37401  finxp3o  37402  poimirlem3  37631  poimirlem5  37633  poimirlem10  37638  poimirlem22  37650  poimirlem23  37651  mendvscafval  43203  binomcxplemnn0  44373  itscnhlinecirc02plem3  48710  inlinecirc02p  48713