Theorem abrexex2 2915

Description: Existence of an existentially restricted class abstraction. φ is normally has free-variable parameters x and y. This is a powerful generalization of abrexex 2912.

Hypotheses

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

Assertion

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

Distinct variable group(s):   x,y,A

Proof of Theorem abrexex2

Step	Hyp	Ref	Expression
1		ax-17 925	. . . 4 ⊢ (∃x ∈ A φ → ∀z∃x ∈ A φ)
2		ax-17 925	. . . . 5 ⊢ (x ∈ A → ∀y x ∈ A)
3		hbs1 986	. . . . 5 ⊢ ([z / y]φ → ∀y[z / y]φ)
4	2, 3	hbrex 1238	. . . 4 ⊢ (∃x ∈ A [z / y]φ → ∀y∃x ∈ A [z / y]φ)
5		sbequ12 865	. . . . 5 ⊢ (y = z → (φ ↔ [z / y]φ))
6	5	birexdv 1220	. . . 4 ⊢ (y = z → (∃x ∈ A φ ↔ ∃x ∈ A [z / y]φ))
7	1, 4, 6	cbvab 1423	. . 3 ⊢ {y∣∃x ∈ A φ} = {z∣∃x ∈ A [z / y]φ}
8		df-clab 1093	. . . . 5 ⊢ (z ∈ {y∣φ} ↔ [z / y]φ)
9	8	birex 1224	. . . 4 ⊢ (∃x ∈ A z ∈ {y∣φ} ↔ ∃x ∈ A [z / y]φ)
10	9	biabi 1181	. . 3 ⊢ {z∣∃x ∈ A z ∈ {y∣φ}} = {z∣∃x ∈ A [z / y]φ}
11	7, 10	eqtr4 1122	. 2 ⊢ {y∣∃x ∈ A φ} = {z∣∃x ∈ A z ∈ {y∣φ}}
12		df-iun 1996	. . 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 2914	. . 3 ⊢ ∪x ∈ A {y∣φ} ∈ V
16	12, 15	eqeltrr 1160	. 2 ⊢ {z∣∃x ∈ A z ∈ {y∣φ}} ∈ V
17	11, 16	eqeltr 1159	1 ⊢ {y∣∃x ∈ A φ} ∈ V

Syntax hints:   = weq 797  [wsb 852  {cab 1090   ∈ wcel 1092  ∃wrex 1202  Vcvv 1348  ∪ciun 1994

This theorem is referenced by:  infxpidmlem9 4941

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-iun 1996  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fv 2438

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