Theorem xpeq12i 5691

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 5688	. 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 5661

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 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-opab 5176  df-xp 5669

This theorem is used by:  imainrect  6181  cnvrescnv  6196  cnvssrndm  6275  fpar  8117  ttrclexg  9699  canthwelem  10650  trclublem  15056  degenmgmnfn  19036  pjpm  21908  txbasval  23814  hausdiag  23853  ussval  24467  ex-xp  30858  hh0oi  32326  fcnvgreu  33088  sitgclg  34797  sitmcl  34806  ismgmOLD  38559  isdrngo1  38665  trrelsuperrel2dg  44455  intxp  49667  isofval2  49867  oppc1stf  50123  oppc2ndf  50124