Theorem xpeq12i 5653

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 5650	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))
4	1, 2, 3	mp2an 693	1 ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐷)

Syntax hints:   = wceq 1542   × cxp 5623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-opab 5149  df-xp 5631

This theorem is referenced by:  imainrect  6140  cnvrescnv  6154  cnvssrndm  6230  fpar  8060  ttrclexg  9638  canthwelem  10567  trclublem  14951  pjpm  21701  txbasval  23584  hausdiag  23623  ussval  24237  ex-xp  30524  hh0oi  31992  fcnvgreu  32763  sitgclg  34505  sitmcl  34514  ismgmOLD  38188  isdrngo1  38294  trrelsuperrel2dg  44119  intxp  49322  isofval2  49522  oppc1stf  49778  oppc2ndf  49779