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Theorem xpriindim 4589
 Description: Distributive law for cross product over relativized indexed intersection. (Contributed by Jim Kingdon, 7-Dec-2018.)
Assertion
Ref Expression
xpriindim (∃𝑦 𝑦𝐴 → (𝐶 × (𝐷 𝑥𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ 𝑥𝐴 (𝐶 × 𝐵)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐶,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦)   𝐷(𝑥,𝑦)

Proof of Theorem xpriindim
StepHypRef Expression
1 xpindi 4586 . 2 (𝐶 × (𝐷 𝑥𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ (𝐶 × 𝑥𝐴 𝐵))
2 xpiindim 4588 . . 3 (∃𝑦 𝑦𝐴 → (𝐶 × 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶 × 𝐵))
32ineq2d 3204 . 2 (∃𝑦 𝑦𝐴 → ((𝐶 × 𝐷) ∩ (𝐶 × 𝑥𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ 𝑥𝐴 (𝐶 × 𝐵)))
41, 3syl5eq 2133 1 (∃𝑦 𝑦𝐴 → (𝐶 × (𝐷 𝑥𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ 𝑥𝐴 (𝐶 × 𝐵)))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1290  ∃wex 1427   ∈ wcel 1439   ∩ cin 3001  ∩ ciin 3739   × cxp 4452 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047 This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2624  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-iin 3741  df-opab 3908  df-xp 4460  df-rel 4461 This theorem is referenced by: (None)
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