Theorem xpeq12i 5647

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 5644	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))
4	1, 2, 3	mp2an 692	1 ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐷)

Syntax hints:   = wceq 1540   × cxp 5617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-opab 5155  df-xp 5625

This theorem is referenced by:  imainrect  6130  cnvrescnv  6144  cnvssrndm  6219  fpar  8049  ttrclexg  9619  canthwelem  10544  trclublem  14902  pjpm  21615  txbasval  23491  hausdiag  23530  ussval  24145  ex-xp  30380  hh0oi  31847  fcnvgreu  32616  sitgclg  34310  sitmcl  34319  ismgmOLD  37830  isdrngo1  37936  trrelsuperrel2dg  43644  intxp  48816  isofval2  49017  oppc1stf  49273  oppc2ndf  49274