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 1569   × cxp 5649

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-opab 5167  df-xp 5657

This theorem is used by:  imainrect  6175  cnvrescnv  6190  cnvssrndm  6276  fpar  8133  ttrclexg  9731  canthwelem  10742  trclublem  15157  degenmgmnfn  19145  pjpm  22023  txbasval  23934  hausdiag  23973  ussval  24587  ex-xp  31046  hh0oi  32514  fcnvgreu  33317  sitgclg  35024  sitmcl  35033  ismgmOLD  38893  isdrngo1  38999  trrelsuperrel2dg  44780  intxpd  50051  isofval2  50254  oppc1stf  50518  oppc2ndf  50519