Theorem elunif 42008

Description: A version of eluni 4799 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 4799	. 2 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵))
2		elunif.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3		nfcv 2920	. . . . 5 ⊢ Ⅎ𝑥𝑦
4	2, 3	nfel 2934	. . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑦
5		elunif.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
6	3, 5	nfel 2934	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
7	4, 6	nfan 1901	. . 3 ⊢ Ⅎ𝑥(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)
8		nfv 1916	. . 3 ⊢ Ⅎ𝑦(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)
9		eleq2w 2836	. . . 4 ⊢ (𝑦 = 𝑥 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑥))
10		eleq1w 2835	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
11	9, 10	anbi12d 634	. . 3 ⊢ (𝑦 = 𝑥 → ((𝐴 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ (𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)))
12	7, 8, 11	cbvexv1 2352	. 2 ⊢ (∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵))
13	1, 12	bitri 278	1 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵))

Syntax hints:   ↔ wb 209   ∧ wa 400  ∃wex 1782   ∈ wcel 2112  Ⅎwnfc 2900  ∪ cuni 4796

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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-v 3412  df-uni 4797

This theorem is referenced by:  eluni2f  42102  stoweidlem46  43044  stoweidlem57  43055