Theorem xpeq12i 5689

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 5686	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))
4	1, 2, 3	mp2an 704	1 ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐷)

Syntax hints:   = wceq 1570   × cxp 5659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-opab 5174  df-xp 5667

This theorem is referenced by:  imainrect  6179  cnvrescnv  6194  cnvssrndm  6272  fpar  8107  ttrclexg  9688  canthwelem  10630  trclublem  15028  pjpm  21858  txbasval  23763  hausdiag  23802  ussval  24416  ex-xp  30787  hh0oi  32255  fcnvgreu  33017  sitgclg  34732  sitmcl  34741  ismgmOLD  38501  isdrngo1  38607  trrelsuperrel2dg  44397  intxp  49610  isofval2  49810  oppc1stf  50066  oppc2ndf  50067