Theorem xpeq12i 5617

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 5614	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))
4	1, 2, 3	mp2an 689	1 ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐷)

Syntax hints:   = wceq 1542   × cxp 5587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-opab 5142  df-xp 5595

This theorem is referenced by:  imainrect  6082  cnvrescnv  6096  cnvssrndm  6172  fpar  7945  ttrclexg  9457  canthwelem  10405  trclublem  14702  pjpm  20911  txbasval  22753  hausdiag  22792  ussval  23407  ex-xp  28794  hh0oi  30259  fcnvgreu  31004  sitgclg  32303  sitmcl  32312  ismgmOLD  36002  isdrngo1  36108  trrelsuperrel2dg  41247