Theorem eluniab 4925

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 4914	. 2 ⊢ (𝐴 ∈ ∪ {𝑥 ∣ 𝜑} ↔ ∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑}))
2		nfv 1911	. . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑦
3		nfsab1 2719	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}
4	2, 3	nfan 1896	. . 3 ⊢ Ⅎ𝑥(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑})
5		nfv 1911	. . 3 ⊢ Ⅎ𝑦(𝐴 ∈ 𝑥 ∧ 𝜑)
6		eleq2w 2822	. . . 4 ⊢ (𝑦 = 𝑥 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑥))
7		eleq1w 2821	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}))
8		abid 2715	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
9	7, 8	bitrdi 287	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑))
10	6, 9	anbi12d 632	. . 3 ⊢ (𝑦 = 𝑥 → ((𝐴 ∈ 𝑦 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑}) ↔ (𝐴 ∈ 𝑥 ∧ 𝜑)))
11	4, 5, 10	cbvexv1 2342	. 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 1775   ∈ wcel 2105  {cab 2711  ∪ cuni 4911

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-v 3479  df-uni 4912

This theorem is referenced by:  elunirab  4926  dfiun2gOLD  5035  inuni  5355  elfv  6904  unielxp  8050  frrlem8  8316  frrlem10  8318  wfrlem12OLD  8358  tfrlem9  8423  dfac5lem2  10161  fin23lem30  10379  unisngl  23550  metrest  24552  aannenlem2  26385  fpwrelmapffslem  32749  dfiota3  35904  mptsnunlem  37320  nnoeomeqom  43301