MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iindif1 Structured version   Visualization version   GIF version

Theorem iindif1 5079
Description: Indexed intersection of class difference with the subtrahend held constant. (Contributed by Thierry Arnoux, 21-Aug-2023.)
Assertion
Ref Expression
iindif1 (𝐴 ≠ ∅ → 𝑥𝐴 (𝐵𝐶) = ( 𝑥𝐴 𝐵𝐶))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem iindif1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 r19.27zv 4511 . . . 4 (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝑦𝐵 ∧ ¬ 𝑦𝐶) ↔ (∀𝑥𝐴 𝑦𝐵 ∧ ¬ 𝑦𝐶)))
2 eldif 3972 . . . . 5 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦𝐶))
32ralbii 3090 . . . 4 (∀𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ ∀𝑥𝐴 (𝑦𝐵 ∧ ¬ 𝑦𝐶))
4 eliin 5000 . . . . . 6 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
54elv 3482 . . . . 5 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
65anbi1i 624 . . . 4 ((𝑦 𝑥𝐴 𝐵 ∧ ¬ 𝑦𝐶) ↔ (∀𝑥𝐴 𝑦𝐵 ∧ ¬ 𝑦𝐶))
71, 3, 63bitr4g 314 . . 3 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ (𝑦 𝑥𝐴 𝐵 ∧ ¬ 𝑦𝐶)))
8 eliin 5000 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∀𝑥𝐴 𝑦 ∈ (𝐵𝐶)))
98elv 3482 . . 3 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∀𝑥𝐴 𝑦 ∈ (𝐵𝐶))
10 eldif 3972 . . 3 (𝑦 ∈ ( 𝑥𝐴 𝐵𝐶) ↔ (𝑦 𝑥𝐴 𝐵 ∧ ¬ 𝑦𝐶))
117, 9, 103bitr4g 314 . 2 (𝐴 ≠ ∅ → (𝑦 𝑥𝐴 (𝐵𝐶) ↔ 𝑦 ∈ ( 𝑥𝐴 𝐵𝐶)))
1211eqrdv 2732 1 (𝐴 ≠ ∅ → 𝑥𝐴 (𝐵𝐶) = ( 𝑥𝐴 𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wne 2937  wral 3058  Vcvv 3477  cdif 3959  c0 4338   ciin 4996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-12 2174  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-v 3479  df-dif 3965  df-nul 4339  df-iin 4998
This theorem is referenced by:  subdrgint  20820
  Copyright terms: Public domain W3C validator