Theorem xpeq12i 5646

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 5643	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))
4	1, 2, 3	mp2an 698	1 ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐷)

Syntax hints:   = wceq 1547   × cxp 5616

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-opab 5135  df-xp 5624

This theorem is referenced by:  imainrect  6132  cnvrescnv  6146  cnvssrndm  6222  fpar  8055  ttrclexg  9635  canthwelem  10564  trclublem  14948  pjpm  21683  txbasval  23589  hausdiag  23628  ussval  24242  ex-xp  30524  hh0oi  31992  fcnvgreu  32764  sitgclg  34526  sitmcl  34535  ismgmOLD  38217  isdrngo1  38323  trrelsuperrel2dg  44115  intxp  49322  isofval2  49522  oppc1stf  49778  oppc2ndf  49779