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Theorem iundif1 35759
Description: Indexed union of class difference with the subtrahend held constant. (Contributed by Brendan Leahy, 6-Aug-2018.)
Assertion
Ref Expression
iundif1 𝑥𝐴 (𝐵𝐶) = ( 𝑥𝐴 𝐵𝐶)
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem iundif1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 r19.41v 3274 . . . 4 (∃𝑥𝐴 (𝑦𝐵 ∧ ¬ 𝑦𝐶) ↔ (∃𝑥𝐴 𝑦𝐵 ∧ ¬ 𝑦𝐶))
2 eldif 3896 . . . . 5 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦𝐶))
32rexbii 3179 . . . 4 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ ∃𝑥𝐴 (𝑦𝐵 ∧ ¬ 𝑦𝐶))
4 eliun 4928 . . . . 5 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
54anbi1i 624 . . . 4 ((𝑦 𝑥𝐴 𝐵 ∧ ¬ 𝑦𝐶) ↔ (∃𝑥𝐴 𝑦𝐵 ∧ ¬ 𝑦𝐶))
61, 3, 53bitr4i 303 . . 3 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ (𝑦 𝑥𝐴 𝐵 ∧ ¬ 𝑦𝐶))
7 eliun 4928 . . 3 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))
8 eldif 3896 . . 3 (𝑦 ∈ ( 𝑥𝐴 𝐵𝐶) ↔ (𝑦 𝑥𝐴 𝐵 ∧ ¬ 𝑦𝐶))
96, 7, 83bitr4i 303 . 2 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ 𝑦 ∈ ( 𝑥𝐴 𝐵𝐶))
109eqriv 2735 1 𝑥𝐴 (𝐵𝐶) = ( 𝑥𝐴 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396   = wceq 1539  wcel 2106  wrex 3065  cdif 3883   ciun 4924
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-v 3431  df-dif 3889  df-iun 4926
This theorem is referenced by: (None)
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