Theorem abrexex2 5771

Description: Existence of an existentially restricted class abstraction. 𝜑 is normally has free-variable parameters 𝑥 and 𝑦. See also abrexex 5764. (Contributed by NM, 12-Sep-2004.)

Hypotheses

Ref	Expression
abrexex2.1	⊢ 𝐴 ∈ V
abrexex2.2	⊢ {𝑦 ∣ 𝜑} ∈ V

Assertion

Ref	Expression
abrexex2	⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V

Distinct variable group:   𝑥,𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem abrexex2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1461	. . . 4 ⊢ Ⅎ𝑧∃𝑥 ∈ 𝐴 𝜑
2		nfcv 2219	. . . . 5 ⊢ Ⅎ𝑦𝐴
3		nfs1v 1856	. . . . 5 ⊢ Ⅎ𝑦[𝑧 / 𝑦]𝜑
4	2, 3	nfrexxy 2403	. . . 4 ⊢ Ⅎ𝑦∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑
5		sbequ12 1694	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑦]𝜑))
6	5	rexbidv 2369	. . . 4 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑))
7	1, 4, 6	cbvab 2201	. . 3 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑}
8		df-clab 2068	. . . . 5 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑧 / 𝑦]𝜑)
9	8	rexbii 2373	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ {𝑦 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑)
10	9	abbii 2194	. . 3 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ {𝑦 ∣ 𝜑}} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑}
11	7, 10	eqtr4i 2104	. 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ {𝑦 ∣ 𝜑}}
12		df-iun 3680	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ {𝑦 ∣ 𝜑}}
13		abrexex2.1	. . . 4 ⊢ 𝐴 ∈ V
14		abrexex2.2	. . . 4 ⊢ {𝑦 ∣ 𝜑} ∈ V
15	13, 14	iunex 5770	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ∈ V
16	12, 15	eqeltrri 2152	. 2 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ {𝑦 ∣ 𝜑}} ∈ V
17	11, 16	eqeltri 2151	1 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V

Syntax hints:   ∈ wcel 1433  [wsb 1685  {cab 2067  ∃wrex 2349  Vcvv 2601  ∪ ciun 3678

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930

This theorem is referenced by:  abexssex  5772  abexex  5773  oprabrexex2  5777  ab2rexex  5778  ab2rexex2  5779