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Theorem iindif1 4960
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 4392 . . . 4 (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝑦𝐵 ∧ ¬ 𝑦𝐶) ↔ (∀𝑥𝐴 𝑦𝐵 ∧ ¬ 𝑦𝐶)))
2 eldif 3853 . . . . 5 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦𝐶))
32ralbii 3080 . . . 4 (∀𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ ∀𝑥𝐴 (𝑦𝐵 ∧ ¬ 𝑦𝐶))
4 eliin 4886 . . . . . 6 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
54elv 3404 . . . . 5 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
65anbi1i 627 . . . 4 ((𝑦 𝑥𝐴 𝐵 ∧ ¬ 𝑦𝐶) ↔ (∀𝑥𝐴 𝑦𝐵 ∧ ¬ 𝑦𝐶))
71, 3, 63bitr4g 317 . . 3 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ (𝑦 𝑥𝐴 𝐵 ∧ ¬ 𝑦𝐶)))
8 eliin 4886 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∀𝑥𝐴 𝑦 ∈ (𝐵𝐶)))
98elv 3404 . . 3 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∀𝑥𝐴 𝑦 ∈ (𝐵𝐶))
10 eldif 3853 . . 3 (𝑦 ∈ ( 𝑥𝐴 𝐵𝐶) ↔ (𝑦 𝑥𝐴 𝐵 ∧ ¬ 𝑦𝐶))
117, 9, 103bitr4g 317 . 2 (𝐴 ≠ ∅ → (𝑦 𝑥𝐴 (𝐵𝐶) ↔ 𝑦 ∈ ( 𝑥𝐴 𝐵𝐶)))
1211eqrdv 2736 1 (𝐴 ≠ ∅ → 𝑥𝐴 (𝐵𝐶) = ( 𝑥𝐴 𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1542  wcel 2114  wne 2934  wral 3053  Vcvv 3398  cdif 3840  c0 4211   ciin 4882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-12 2179  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ne 2935  df-ral 3058  df-v 3400  df-dif 3846  df-nul 4212  df-iin 4884
This theorem is referenced by:  subdrgint  19701
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