Theorem eluniab 4865

Description: Membership in union of a class abstraction. (Contributed by NM, 11-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2016.)

Assertion

Ref	Expression
eluniab	⊢ (𝐴 ∈ ∪ {𝑥 ∣ 𝜑} ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝜑))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem eluniab

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluni 4854	. 2 ⊢ (𝐴 ∈ ∪ {𝑥 ∣ 𝜑} ↔ ∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑}))
2		nfv 1916	. . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑦
3		nfsab1 2723	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}
4	2, 3	nfan 1901	. . 3 ⊢ Ⅎ𝑥(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑})
5		nfv 1916	. . 3 ⊢ Ⅎ𝑦(𝐴 ∈ 𝑥 ∧ 𝜑)
6		eleq2w 2821	. . . 4 ⊢ (𝑦 = 𝑥 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑥))
7		eleq1w 2820	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}))
8		abid 2719	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
9	7, 8	bitrdi 287	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑))
10	6, 9	anbi12d 633	. . 3 ⊢ (𝑦 = 𝑥 → ((𝐴 ∈ 𝑦 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑}) ↔ (𝐴 ∈ 𝑥 ∧ 𝜑)))
11	4, 5, 10	cbvexv1 2347	. 2 ⊢ (∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑}) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝜑))
12	1, 11	bitri 275	1 ⊢ (𝐴 ∈ ∪ {𝑥 ∣ 𝜑} ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝜑))

Syntax hints:   ↔ wb 206   ∧ wa 395  ∃wex 1781   ∈ wcel 2114  {cab 2715  ∪ cuni 4851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3432  df-uni 4852

This theorem is referenced by:  elunirab  4866  inuni  5287  elfv  6832  unielxp  7973  frrlem8  8236  frrlem10  8238  tfrlem9  8317  dfac5lem2  10037  fin23lem30  10255  unisngl  23502  metrest  24499  aannenlem2  26306  fpwrelmapffslem  32820  dfiota3  36119  mptsnunlem  37668  nnoeomeqom  43758