Theorem repizf2 3906

Description: Replacement. This version of replacement is stronger than repizf 3864 in the sense that φ does not need to map all values of x in w to a value of y. The resulting set contains those elements for which there is a value of y and in that sense, this theorem combines repizf 3864 with ax-sep 3866. Another variation would be ∀x ∈ w∃*yφ → {y ∣ ∃x(x ∈ w ∧ φ)} ∈ V but we don't have a proof of that yet. (Contributed by Jim Kingdon, 7-Sep-2018.)

Hypothesis

Ref	Expression
repizf2.1	⊢ Ⅎzφ

Assertion

Ref	Expression
repizf2	⊢ (∀x ∈ w ∃*yφ → ∃z∀x ∈ {x ∈ w ∣ ∃yφ}∃y ∈ z φ)

Distinct variable group:   x,y,z,w

Allowed substitution hints:   φ(x,y,z,w)

Proof of Theorem repizf2

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2554	. . 3 ⊢ w ∈ V
2	1	rabex 3892	. 2 ⊢ {x ∈ w ∣ ∃yφ} ∈ V
3		repizf2lem 3905	. . . 4 ⊢ (∀x ∈ w ∃*yφ ↔ ∀x ∈ {x ∈ w ∣ ∃yφ}∃!yφ)
4		nfcv 2175	. . . . . 6 ⊢ Ⅎxv
5		nfrab1 2483	. . . . . 6 ⊢ Ⅎx{x ∈ w ∣ ∃yφ}
6	4, 5	raleqf 2495	. . . . 5 ⊢ (v = {x ∈ w ∣ ∃yφ} → (∀x ∈ v ∃!yφ ↔ ∀x ∈ {x ∈ w ∣ ∃yφ}∃!yφ))
7		repizf2.1	. . . . . 6 ⊢ Ⅎzφ
8	7	repizf 3864	. . . . 5 ⊢ (∀x ∈ v ∃!yφ → ∃z∀x ∈ v ∃y ∈ z φ)
9	6, 8	syl6bir 153	. . . 4 ⊢ (v = {x ∈ w ∣ ∃yφ} → (∀x ∈ {x ∈ w ∣ ∃yφ}∃!yφ → ∃z∀x ∈ v ∃y ∈ z φ))
10	3, 9	syl5bi 141	. . 3 ⊢ (v = {x ∈ w ∣ ∃yφ} → (∀x ∈ w ∃*yφ → ∃z∀x ∈ v ∃y ∈ z φ))
11		df-rab 2309	. . . . . 6 ⊢ {x ∈ w ∣ ∃yφ} = {x ∣ (x ∈ w ∧ ∃yφ)}
12		nfv 1418	. . . . . . . 8 ⊢ Ⅎz x ∈ w
13	7	nfex 1525	. . . . . . . 8 ⊢ Ⅎz∃yφ
14	12, 13	nfan 1454	. . . . . . 7 ⊢ Ⅎz(x ∈ w ∧ ∃yφ)
15	14	nfab 2179	. . . . . 6 ⊢ Ⅎz{x ∣ (x ∈ w ∧ ∃yφ)}
16	11, 15	nfcxfr 2172	. . . . 5 ⊢ Ⅎz{x ∈ w ∣ ∃yφ}
17	16	nfeq2 2186	. . . 4 ⊢ Ⅎz v = {x ∈ w ∣ ∃yφ}
18	4, 5	raleqf 2495	. . . 4 ⊢ (v = {x ∈ w ∣ ∃yφ} → (∀x ∈ v ∃y ∈ z φ ↔ ∀x ∈ {x ∈ w ∣ ∃yφ}∃y ∈ z φ))
19	17, 18	exbid 1504	. . 3 ⊢ (v = {x ∈ w ∣ ∃yφ} → (∃z∀x ∈ v ∃y ∈ z φ ↔ ∃z∀x ∈ {x ∈ w ∣ ∃yφ}∃y ∈ z φ))
20	10, 19	sylibd 138	. 2 ⊢ (v = {x ∈ w ∣ ∃yφ} → (∀x ∈ w ∃*yφ → ∃z∀x ∈ {x ∈ w ∣ ∃yφ}∃y ∈ z φ))
21	2, 20	vtocle 2621	1 ⊢ (∀x ∈ w ∃*yφ → ∃z∀x ∈ {x ∈ w ∣ ∃yφ}∃y ∈ z φ)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242  Ⅎwnf 1346  ∃wex 1378  ∃!weu 1897  ∃*wmo 1898  {cab 2023  ∀wral 2300  ∃wrex 2301  {crab 2304

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-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866

This theorem depends on definitions:  df-bi 110  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-rab 2309  df-v 2553  df-in 2918  df-ss 2925

This theorem is referenced by: (None)