Theorem xpeq12 5703

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 5692	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶))
2		xpeq2 5699	. 2 ⊢ (𝐶 = 𝐷 → (𝐵 × 𝐶) = (𝐵 × 𝐷))
3	1, 2	sylan9eq 2785	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   × cxp 5676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-opab 5212  df-xp 5684

This theorem is referenced by:  xpeq12i  5706  xpeq12d  5709  xpid11  5934  xp11  6181  infxpenlem  10038  fpwwe2lem4  10659  pwfseqlem4a  10686  pwfseqlem4  10687  pwfseqlem5  10688  pwfseq  10689  pwsval  17471  mamufval  22336  mvmulfval  22488  txtopon  23539  txbasval  23554  txindislem  23581  ismet  24273  isxmet  24274  shsval  31194  sat1el2xp  35117  bj-imdirvallem  36787  prdsbnd2  37396  ismgmOLD  37451  opidon2OLD  37455  ttac  42596  rfovd  43570  fsovrfovd  43578  sblpnf  43886