Theorem xpeq12 5636

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 5625	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶))
2		xpeq2 5632	. 2 ⊢ (𝐶 = 𝐷 → (𝐵 × 𝐶) = (𝐵 × 𝐷))
3	1, 2	sylan9eq 2786	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   × cxp 5609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-opab 5149  df-xp 5617

This theorem is referenced by:  xpeq12i  5639  xpeq12d  5642  xpid11  5867  xp11  6117  infxpenlem  9899  pwfseqlem4a  10547  pwfseqlem4  10548  pwfseqlem5  10549  pwfseq  10550  pwsval  17385  mamufval  22302  mvmulfval  22452  txtopon  23501  txbasval  23516  txindislem  23543  ismet  24233  isxmet  24234  shsval  31284  sat1el2xp  35415  bj-imdirvallem  37214  prdsbnd2  37835  ismgmOLD  37890  opidon2OLD  37894  ttac  43069  rfovd  44034  fsovrfovd  44042  sblpnf  44343