Theorem iundif1 32553

Description: Indexed union of class difference with the subtrahend held constant. (Contributed by Brendan Leahy, 6-Aug-2018.)

Assertion

Ref	Expression
iundif1	⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) = (∪ 𝑥 ∈ 𝐴 𝐵 ∖ 𝐶)

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.

Step	Hyp	Ref	Expression
1		r19.41v 3070	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶))
2		eldif 3550	. . . . 5 ⊢ (𝑦 ∈ (𝐵 ∖ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶))
3	2	rexbii 3023	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶) ↔ ∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶))
4		eliun 4460	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
5	4	anbi1i 727	. . . 4 ⊢ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ ¬ 𝑦 ∈ 𝐶) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶))
6	1, 3, 5	3bitr4i 291	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶) ↔ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ ¬ 𝑦 ∈ 𝐶))
7		eliun 4460	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶))
8		eldif 3550	. . 3 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ 𝐶) ↔ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ ¬ 𝑦 ∈ 𝐶))
9	6, 7, 8	3bitr4i 291	. 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) ↔ 𝑦 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ 𝐶))
10	9	eqriv 2607	1 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) = (∪ 𝑥 ∈ 𝐴 𝐵 ∖ 𝐶)

Syntax hints:  ¬ wn 3   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   ∖ cdif 3537  ∪ ciun 4455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-iun 4457

This theorem is referenced by: (None)