Theorem xpeq12i 5669

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 5666	. 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 5639

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 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-opab 5173  df-xp 5647

This theorem is referenced by:  imainrect  6157  cnvrescnv  6171  cnvssrndm  6247  fpar  8098  ttrclexg  9683  canthwelem  10610  trclublem  14968  pjpm  21624  txbasval  23500  hausdiag  23539  ussval  24154  ex-xp  30372  hh0oi  31839  fcnvgreu  32604  sitgclg  34340  sitmcl  34349  ismgmOLD  37851  isdrngo1  37957  trrelsuperrel2dg  43667  intxp  48824  isofval2  49025  oppc1stf  49281  oppc2ndf  49282