Theorem xpeq12 5639

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 5628	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶))
2		xpeq2 5635	. 2 ⊢ (𝐶 = 𝐷 → (𝐵 × 𝐶) = (𝐵 × 𝐷))
3	1, 2	sylan9eq 2796	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1540   × cxp 5612

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-opab 5152  df-xp 5620

This theorem is referenced by:  xpeq12i  5642  xpeq12d  5645  xpid11  5867  xp11  6107  infxpenlem  9862  fpwwe2lem4  10483  pwfseqlem4a  10510  pwfseqlem4  10511  pwfseqlem5  10512  pwfseq  10513  pwsval  17286  mamufval  21632  mvmulfval  21789  txtopon  22840  txbasval  22855  txindislem  22882  ismet  23574  isxmet  23575  shsval  29903  sat1el2xp  33581  bj-imdirvallem  35449  prdsbnd2  36051  ismgmOLD  36106  opidon2OLD  36110  ttac  41109  rfovd  41919  fsovrfovd  41927  sblpnf  42238