Theorem xpeq12i 5679

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 5676	. 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 5649

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 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-opab 5168  df-xp 5657

This theorem is used by:  imainrect  6173  cnvrescnv  6188  cnvssrndm  6272  fpar  8125  ttrclexg  9717  canthwelem  10728  trclublem  15141  degenmgmnfn  19129  pjpm  22007  txbasval  23918  hausdiag  23957  ussval  24571  ex-xp  31030  hh0oi  32498  fcnvgreu  33259  sitgclg  34967  sitmcl  34976  ismgmOLD  38764  isdrngo1  38870  trrelsuperrel2dg  44656  intxpd  49911  isofval2  50109  oppc1stf  50365  oppc2ndf  50366