Theorem xpeq12 5702

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 5691	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶))
2		xpeq2 5698	. 2 ⊢ (𝐶 = 𝐷 → (𝐵 × 𝐶) = (𝐵 × 𝐷))
3	1, 2	sylan9eq 2793	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   × cxp 5675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-opab 5212  df-xp 5683

This theorem is referenced by:  xpeq12i  5705  xpeq12d  5708  xpid11  5932  xp11  6175  infxpenlem  10008  fpwwe2lem4  10629  pwfseqlem4a  10656  pwfseqlem4  10657  pwfseqlem5  10658  pwfseq  10659  pwsval  17432  mamufval  21887  mvmulfval  22044  txtopon  23095  txbasval  23110  txindislem  23137  ismet  23829  isxmet  23830  shsval  30565  sat1el2xp  34370  bj-imdirvallem  36061  prdsbnd2  36663  ismgmOLD  36718  opidon2OLD  36722  ttac  41775  rfovd  42752  fsovrfovd  42760  sblpnf  43069