Theorem xpeq12i 5650

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 5647	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))
4	1, 2, 3	mp2an 692	1 ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐷)

Syntax hints:   = wceq 1541   × cxp 5620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-opab 5159  df-xp 5628

This theorem is referenced by:  imainrect  6137  cnvrescnv  6151  cnvssrndm  6227  fpar  8056  ttrclexg  9630  canthwelem  10559  trclublem  14916  pjpm  21661  txbasval  23548  hausdiag  23587  ussval  24201  ex-xp  30460  hh0oi  31927  fcnvgreu  32700  sitgclg  34448  sitmcl  34457  ismgmOLD  37990  isdrngo1  38096  trrelsuperrel2dg  43854  intxp  49019  isofval2  49219  oppc1stf  49475  oppc2ndf  49476