Theorem xpeq12i 5686

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 5683	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))
4	1, 2, 3	mp2an 705	1 ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐷)

This proof depends on syntax axioms:   = wceq 1570   × cxp 5656

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-opab 5174  df-xp 5664

This theorem is used by:  imainrect  6191  cnvrescnv  6206  cnvssrndm  6297  fpar  8164  ttrclexg  9761  canthwelem  10774  trclublem  15191  degenmgmnfn  19186  pjpm  22069  txbasval  23980  hausdiag  24019  ussval  24633  ex-xp  31094  hh0oi  32562  fcnvgreu  33364  sitgclg  35085  sitmcl  35094  ismgmOLD  39050  isdrngo1  39156  trrelsuperrel2dg  44932  intxpd  50274  isofval2  50477  oppc1stf  50741  oppc2ndf  50742