Theorem xpeq12 5168

Description: Equality theorem for Cartesian product. (Contributed by FL, 31-Aug-2009.)

Assertion

Ref	Expression
xpeq12	⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))

Proof of Theorem xpeq12

Step	Hyp	Ref	Expression
1		xpeq1 5157	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶))
2		xpeq2 5163	. 2 ⊢ (𝐶 = 𝐷 → (𝐵 × 𝐶) = (𝐵 × 𝐷))
3	1, 2	sylan9eq 2705	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   × cxp 5141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-opab 4746  df-xp 5149

This theorem is referenced by:  xpeq12i  5171  xpeq12d  5174  xpid11  5379  xp11  5604  infxpenlem  8874  fpwwe2lem5  9494  pwfseqlem4a  9521  pwfseqlem4  9522  pwfseqlem5  9523  pwfseq  9524  pwsval  16193  mamufval  20239  mvmulfval  20396  txtopon  21442  txbasval  21457  txindislem  21484  ismet  22175  isxmet  22176  shsval  28299  prdsbnd2  33724  ismgmOLD  33779  opidon2OLD  33783  ttac  37920  rfovd  38612  fsovrfovd  38620  sblpnf  38826