Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  iinin1 Structured version   Visualization version   GIF version

Theorem iinin1 4731
 Description: Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4714 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.)
Assertion
Ref Expression
iinin1 (𝐴 ≠ ∅ → 𝑥𝐴 (𝐶𝐵) = ( 𝑥𝐴 𝐶𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem iinin1
StepHypRef Expression
1 iinin2 4730 . 2 (𝐴 ≠ ∅ → 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶))
2 incom 3936 . . . 4 (𝐶𝐵) = (𝐵𝐶)
32a1i 11 . . 3 (𝑥𝐴 → (𝐶𝐵) = (𝐵𝐶))
43iineq2i 4680 . 2 𝑥𝐴 (𝐶𝐵) = 𝑥𝐴 (𝐵𝐶)
5 incom 3936 . 2 ( 𝑥𝐴 𝐶𝐵) = (𝐵 𝑥𝐴 𝐶)
61, 4, 53eqtr4g 2807 1 (𝐴 ≠ ∅ → 𝑥𝐴 (𝐶𝐵) = ( 𝑥𝐴 𝐶𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1620   ∈ wcel 2127   ≠ wne 2920   ∩ cin 3702  ∅c0 4046  ∩ ciin 4661 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-v 3330  df-dif 3706  df-in 3710  df-nul 4047  df-iin 4663 This theorem is referenced by:  firest  16266  iniin1  39777
 Copyright terms: Public domain W3C validator