Theorem xpeq12i 5710

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 5707	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))
4	1, 2, 3	mp2an 690	1 ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐷)

Syntax hints:   = wceq 1534   × cxp 5680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-opab 5216  df-xp 5688

This theorem is referenced by:  imainrect  6192  cnvrescnv  6206  cnvssrndm  6282  fpar  8130  ttrclexg  9766  canthwelem  10693  trclublem  15000  pjpm  21706  txbasval  23601  hausdiag  23640  ussval  24255  ex-xp  30369  hh0oi  31836  fcnvgreu  32590  sitgclg  34176  sitmcl  34185  ismgmOLD  37551  isdrngo1  37657  trrelsuperrel2dg  43338