Theorem elunif 45465

Description: A version of eluni 4848 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 4848	. 2 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵))
2		elunif.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3		nfcv 2902	. . . . 5 ⊢ Ⅎ𝑥𝑦
4	2, 3	nfel 2916	. . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑦
5		elunif.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
6	3, 5	nfel 2916	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
7	4, 6	nfan 1906	. . 3 ⊢ Ⅎ𝑥(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)
8		nfv 1921	. . 3 ⊢ Ⅎ𝑦(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)
9		eleq2w 2824	. . . 4 ⊢ (𝑦 = 𝑥 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑥))
10		eleq1w 2823	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
11	9, 10	anbi12d 638	. . 3 ⊢ (𝑦 = 𝑥 → ((𝐴 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ (𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)))
12	7, 8, 11	cbvexv1 2350	. 2 ⊢ (∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵))
13	1, 12	bitri 276	1 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵))

Syntax hints:   ↔ wb 207   ∧ wa 396  ∃wex 1786   ∈ wcel 2119  Ⅎwnfc 2887  ∪ cuni 4845

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-v 3434  df-uni 4846

This theorem is referenced by:  eluni2f  45551  stoweidlem46  46490  stoweidlem57  46501