Theorem eluniab 4762

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 4754	. 2 ⊢ (𝐴 ∈ ∪ {𝑥 ∣ 𝜑} ↔ ∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑}))
2		nfv 1896	. . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑦
3		nfsab1 2786	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}
4	2, 3	nfan 1885	. . 3 ⊢ Ⅎ𝑥(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑})
5		nfv 1896	. . 3 ⊢ Ⅎ𝑦(𝐴 ∈ 𝑥 ∧ 𝜑)
6		eleq2w 2868	. . . 4 ⊢ (𝑦 = 𝑥 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑥))
7		eleq1w 2867	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}))
8		abid 2781	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
9	7, 8	syl6bb 288	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑))
10	6, 9	anbi12d 630	. . 3 ⊢ (𝑦 = 𝑥 → ((𝐴 ∈ 𝑦 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑}) ↔ (𝐴 ∈ 𝑥 ∧ 𝜑)))
11	4, 5, 10	cbvexv1 2323	. 2 ⊢ (∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑}) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝜑))
12	1, 11	bitri 276	1 ⊢ (𝐴 ∈ ∪ {𝑥 ∣ 𝜑} ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝜑))

Syntax hints:   ↔ wb 207   ∧ wa 396  ∃wex 1765   ∈ wcel 2083  {cab 2777  ∪ cuni 4751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-v 3442  df-uni 4752

This theorem is referenced by:  elunirab  4763  dfiun2g  4864  dfiun2gOLD  4865  inuni  5144  elfv  6543  unielxp  7590  wfrlem12  7825  tfrlem9  7880  dfac5lem2  9403  fin23lem30  9617  unisngl  21823  metrest  22821  aannenlem2  24605  fpwrelmapffslem  30152  frrlem8  32741  frrlem10  32743  dfiota3  32995  mptsnunlem  34171