Theorem xpeq12i 5608

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 5605	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))
4	1, 2, 3	mp2an 688	1 ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐷)

Syntax hints:   = wceq 1539   × cxp 5578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-opab 5133  df-xp 5586

This theorem is referenced by:  imainrect  6073  cnvrescnv  6087  cnvssrndm  6163  fpar  7927  canthwelem  10337  trclublem  14634  pjpm  20825  txbasval  22665  hausdiag  22704  ussval  23319  ex-xp  28701  hh0oi  30166  fcnvgreu  30912  sitgclg  32209  sitmcl  32218  ttrclexg  33709  ismgmOLD  35935  isdrngo1  36041  trrelsuperrel2dg  41168