Theorem xpeq12i 5717

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 5714	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))
4	1, 2, 3	mp2an 692	1 ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐷)

Syntax hints:   = wceq 1537   × cxp 5687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-opab 5211  df-xp 5695

This theorem is referenced by:  imainrect  6203  cnvrescnv  6217  cnvssrndm  6293  fpar  8140  ttrclexg  9761  canthwelem  10688  trclublem  15031  pjpm  21746  txbasval  23630  hausdiag  23669  ussval  24284  ex-xp  30465  hh0oi  31932  fcnvgreu  32690  sitgclg  34324  sitmcl  34333  ismgmOLD  37837  isdrngo1  37943  trrelsuperrel2dg  43661