Theorem xpeq12i 5690

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 5687	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))
4	1, 2, 3	mp2an 704	1 ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐷)

Syntax hints:   = wceq 1567   × cxp 5660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-opab 5178  df-xp 5668

This theorem is referenced by:  imainrect  6180  cnvrescnv  6195  cnvssrndm  6273  fpar  8111  ttrclexg  9692  canthwelem  10635  trclublem  15032  pjpm  21827  txbasval  23732  hausdiag  23771  ussval  24385  ex-xp  30728  hh0oi  32196  fcnvgreu  32958  sitgclg  34677  sitmcl  34686  ismgmOLD  38389  isdrngo1  38495  trrelsuperrel2dg  44289  intxp  49495  isofval2  49695  oppc1stf  49951  oppc2ndf  49952