Theorem xpeq12 5714

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 5703	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶))
2		xpeq2 5710	. 2 ⊢ (𝐶 = 𝐷 → (𝐵 × 𝐶) = (𝐵 × 𝐷))
3	1, 2	sylan9eq 2795	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   × cxp 5687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-opab 5211  df-xp 5695

This theorem is referenced by:  xpeq12i  5717  xpeq12d  5720  xpid11  5946  xp11  6197  infxpenlem  10051  pwfseqlem4a  10699  pwfseqlem4  10700  pwfseqlem5  10701  pwfseq  10702  pwsval  17533  mamufval  22412  mvmulfval  22564  txtopon  23615  txbasval  23630  txindislem  23657  ismet  24349  isxmet  24350  shsval  31341  sat1el2xp  35364  bj-imdirvallem  37163  prdsbnd2  37782  ismgmOLD  37837  opidon2OLD  37841  ttac  43025  rfovd  43991  fsovrfovd  43999  sblpnf  44306