Theorem xpeq12i 5675

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 5672	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))
4	1, 2, 3	mp2an 702	1 ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐷)

Syntax hints:   = wceq 1560   × cxp 5645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-opab 5163  df-xp 5653

This theorem is referenced by:  imainrect  6167  cnvrescnv  6182  cnvssrndm  6258  fpar  8095  ttrclexg  9678  canthwelem  10608  trclublem  15008  pjpm  21757  txbasval  23663  hausdiag  23702  ussval  24316  ex-xp  30635  hh0oi  32103  fcnvgreu  32871  sitgclg  34636  sitmcl  34645  ismgmOLD  38346  isdrngo1  38452  trrelsuperrel2dg  44244  intxp  49450  isofval2  49650  oppc1stf  49906  oppc2ndf  49907