Theorem xpeq12i 5704

Description: Equality inference for Cartesian product. (Contributed by FL, 31-Aug-2009.)

Hypotheses

Ref	Expression
xpeq12i.1	⊢ 𝐴 = 𝐵
xpeq12i.2	⊢ 𝐶 = 𝐷

Assertion

Ref	Expression
xpeq12i	⊢ (𝐴 × 𝐶) = (𝐵 × 𝐷)

Proof of Theorem xpeq12i

Step	Hyp	Ref	Expression
1		xpeq12i.1	. 2 ⊢ 𝐴 = 𝐵
2		xpeq12i.2	. 2 ⊢ 𝐶 = 𝐷
3		xpeq12 5701	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))
4	1, 2, 3	mp2an 690	1 ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐷)

Syntax hints:   = wceq 1541   × cxp 5674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-opab 5211  df-xp 5682

This theorem is referenced by:  imainrect  6180  cnvrescnv  6194  cnvssrndm  6270  fpar  8104  ttrclexg  9720  canthwelem  10647  trclublem  14946  pjpm  21482  txbasval  23330  hausdiag  23369  ussval  23984  ex-xp  29944  hh0oi  31411  fcnvgreu  32153  sitgclg  33627  sitmcl  33636  ismgmOLD  37021  isdrngo1  37127  trrelsuperrel2dg  42724