Theorem elunif 39175

Description: A version of eluni 4439 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
elunif.1	⊢ Ⅎ𝑥𝐴
elunif.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
elunif	⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵))

Distinct variable group:   𝐴,𝐵

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem elunif

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluni 4439	. 2 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵))
2		elunif.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3		nfcv 2764	. . . . 5 ⊢ Ⅎ𝑥𝑦
4	2, 3	nfel 2777	. . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑦
5		elunif.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
6	3, 5	nfel 2777	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
7	4, 6	nfan 1828	. . 3 ⊢ Ⅎ𝑥(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)
8		nfv 1843	. . 3 ⊢ Ⅎ𝑦(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)
9		eleq2 2690	. . . 4 ⊢ (𝑦 = 𝑥 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑥))
10		eleq1 2689	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
11	9, 10	anbi12d 747	. . 3 ⊢ (𝑦 = 𝑥 → ((𝐴 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ (𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)))
12	7, 8, 11	cbvex 2272	. 2 ⊢ (∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵))
13	1, 12	bitri 264	1 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵))

Syntax hints:   ↔ wb 196   ∧ wa 384  ∃wex 1704   ∈ wcel 1990  Ⅎwnfc 2751  ∪ cuni 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-uni 4437

This theorem is referenced by:  eluni2f  39286  stoweidlem46  40263  stoweidlem57  40274