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Theorem iindif1 4973
 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 4427 . . . 4 (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝑦𝐵 ∧ ¬ 𝑦𝐶) ↔ (∀𝑥𝐴 𝑦𝐵 ∧ ¬ 𝑦𝐶)))
2 eldif 3923 . . . . 5 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦𝐶))
32ralbii 3152 . . . 4 (∀𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ ∀𝑥𝐴 (𝑦𝐵 ∧ ¬ 𝑦𝐶))
4 eliin 4900 . . . . . 6 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
54elv 3478 . . . . 5 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
65anbi1i 625 . . . 4 ((𝑦 𝑥𝐴 𝐵 ∧ ¬ 𝑦𝐶) ↔ (∀𝑥𝐴 𝑦𝐵 ∧ ¬ 𝑦𝐶))
71, 3, 63bitr4g 316 . . 3 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ (𝑦 𝑥𝐴 𝐵 ∧ ¬ 𝑦𝐶)))
8 eliin 4900 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∀𝑥𝐴 𝑦 ∈ (𝐵𝐶)))
98elv 3478 . . 3 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∀𝑥𝐴 𝑦 ∈ (𝐵𝐶))
10 eldif 3923 . . 3 (𝑦 ∈ ( 𝑥𝐴 𝐵𝐶) ↔ (𝑦 𝑥𝐴 𝐵 ∧ ¬ 𝑦𝐶))
117, 9, 103bitr4g 316 . 2 (𝐴 ≠ ∅ → (𝑦 𝑥𝐴 (𝐵𝐶) ↔ 𝑦 ∈ ( 𝑥𝐴 𝐵𝐶)))
1211eqrdv 2818 1 (𝐴 ≠ ∅ → 𝑥𝐴 (𝐵𝐶) = ( 𝑥𝐴 𝐵𝐶))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ≠ wne 3006  ∀wral 3125  Vcvv 3473   ∖ cdif 3910  ∅c0 4269  ∩ ciin 4896 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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-v 3475  df-dif 3916  df-nul 4270  df-iin 4898 This theorem is referenced by:  subdrgint  19558
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