Theorem xpeq12i 5683

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 5680	. 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 5653

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-opab 5168  df-xp 5661

This theorem is used by:  imainrect  6174  cnvrescnv  6189  cnvssrndm  6268  fpar  8113  ttrclexg  9702  canthwelem  10659  trclublem  15068  degenmgmnfn  19049  pjpm  21921  txbasval  23832  hausdiag  23871  ussval  24485  ex-xp  30916  hh0oi  32384  fcnvgreu  33145  sitgclg  34853  sitmcl  34862  ismgmOLD  38600  isdrngo1  38706  trrelsuperrel2dg  44511  intxp  49760  isofval2  49958  oppc1stf  50214  oppc2ndf  50215