Theorem abrexex2 5751

Description: Existence of an existentially restricted class abstraction. 𝜑 is normally has free-variable parameters 𝑥 and 𝑦. See also abrexex 5744. (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 1421	. . . 4 ⊢ Ⅎ𝑧∃𝑥 ∈ 𝐴 𝜑
2		nfcv 2178	. . . . 5 ⊢ Ⅎ𝑦𝐴
3		nfs1v 1815	. . . . 5 ⊢ Ⅎ𝑦[𝑧 / 𝑦]𝜑
4	2, 3	nfrexxy 2361	. . . 4 ⊢ Ⅎ𝑦∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑
5		sbequ12 1654	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑦]𝜑))
6	5	rexbidv 2327	. . . 4 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑))
7	1, 4, 6	cbvab 2160	. . 3 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑}
8		df-clab 2027	. . . . 5 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑧 / 𝑦]𝜑)
9	8	rexbii 2331	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ {𝑦 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑)
10	9	abbii 2153	. . 3 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ {𝑦 ∣ 𝜑}} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑}
11	7, 10	eqtr4i 2063	. 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ {𝑦 ∣ 𝜑}}
12		df-iun 3659	. . 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 5750	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ∈ V
16	12, 15	eqeltrri 2111	. 2 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ {𝑦 ∣ 𝜑}} ∈ V
17	11, 16	eqeltri 2110	1 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V

Syntax hints:   ∈ wcel 1393  [wsb 1645  {cab 2026  ∃wrex 2307  Vcvv 2557  ∪ ciun 3657

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910

This theorem is referenced by:  abexssex  5752  abexex  5753  oprabrexex2  5757  ab2rexex  5758  ab2rexex2  5759