Theorem abrexex2 5693

Description: Existence of an existentially restricted class abstraction. φ is normally has free-variable parameters x and y. See also abrexex 5686. (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 1418	. . . 4 ⊢ Ⅎz∃x ∈ A φ
2		nfcv 2175	. . . . 5 ⊢ ℲyA
3		nfs1v 1812	. . . . 5 ⊢ Ⅎy[z / y]φ
4	2, 3	nfrexxy 2355	. . . 4 ⊢ Ⅎy∃x ∈ A [z / y]φ
5		sbequ12 1651	. . . . 5 ⊢ (y = z → (φ ↔ [z / y]φ))
6	5	rexbidv 2321	. . . 4 ⊢ (y = z → (∃x ∈ A φ ↔ ∃x ∈ A [z / y]φ))
7	1, 4, 6	cbvab 2157	. . 3 ⊢ {y ∣ ∃x ∈ A φ} = {z ∣ ∃x ∈ A [z / y]φ}
8		df-clab 2024	. . . . 5 ⊢ (z ∈ {y ∣ φ} ↔ [z / y]φ)
9	8	rexbii 2325	. . . 4 ⊢ (∃x ∈ A z ∈ {y ∣ φ} ↔ ∃x ∈ A [z / y]φ)
10	9	abbii 2150	. . 3 ⊢ {z ∣ ∃x ∈ A z ∈ {y ∣ φ}} = {z ∣ ∃x ∈ A [z / y]φ}
11	7, 10	eqtr4i 2060	. 2 ⊢ {y ∣ ∃x ∈ A φ} = {z ∣ ∃x ∈ A z ∈ {y ∣ φ}}
12		df-iun 3650	. . 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 5692	. . 3 ⊢ ∪ x ∈ A {y ∣ φ} ∈ V
16	12, 15	eqeltrri 2108	. 2 ⊢ {z ∣ ∃x ∈ A z ∈ {y ∣ φ}} ∈ V
17	11, 16	eqeltri 2107	1 ⊢ {y ∣ ∃x ∈ A φ} ∈ V

Syntax hints:   ∈ wcel 1390  [wsb 1642  {cab 2023  ∃wrex 2301  Vcvv 2551  ∪ ciun 3648

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853

This theorem is referenced by:  abexssex  5694  abexex  5695  oprabrexex2  5699  ab2rexex  5700  ab2rexex2  5701