Theorem repizf2 3885

Description: Replacement. This version of replacement is stronger than repizf 3843 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 3843 with ax-sep 3845. 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 2534	. . 3 ⊢ w ∈ V
2	1	rabex 3871	. 2 ⊢ {x ∈ w ∣ ∃yφ} ∈ V
3		repizf2lem 3884	. . . 4 ⊢ (∀x ∈ w ∃*yφ ↔ ∀x ∈ {x ∈ w ∣ ∃yφ}∃!yφ)
4		nfcv 2156	. . . . . 6 ⊢ Ⅎxv
5		nfrab1 2463	. . . . . 6 ⊢ Ⅎx{x ∈ w ∣ ∃yφ}
6	4, 5	raleqf 2475	. . . . 5 ⊢ (v = {x ∈ w ∣ ∃yφ} → (∀x ∈ v ∃!yφ ↔ ∀x ∈ {x ∈ w ∣ ∃yφ}∃!yφ))
7		repizf2.1	. . . . . 6 ⊢ Ⅎzφ
8	7	repizf 3843	. . . . 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 2289	. . . . . 6 ⊢ {x ∈ w ∣ ∃yφ} = {x ∣ (x ∈ w ∧ ∃yφ)}
12		nfv 1398	. . . . . . . 8 ⊢ Ⅎz x ∈ w
13	7	nfex 1506	. . . . . . . 8 ⊢ Ⅎz∃yφ
14	12, 13	nfan 1435	. . . . . . 7 ⊢ Ⅎz(x ∈ w ∧ ∃yφ)
15	14	nfab 2160	. . . . . 6 ⊢ Ⅎz{x ∣ (x ∈ w ∧ ∃yφ)}
16	11, 15	nfcxfr 2153	. . . . 5 ⊢ Ⅎz{x ∈ w ∣ ∃yφ}
17	16	nfeq2 2167	. . . 4 ⊢ Ⅎz v = {x ∈ w ∣ ∃yφ}
18	4, 5	raleqf 2475	. . . 4 ⊢ (v = {x ∈ w ∣ ∃yφ} → (∀x ∈ v ∃y ∈ z φ ↔ ∀x ∈ {x ∈ w ∣ ∃yφ}∃y ∈ z φ))
19	17, 18	exbid 1485	. . 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 2600	1 ⊢ (∀x ∈ w ∃*yφ → ∃z∀x ∈ {x ∈ w ∣ ∃yφ}∃y ∈ z φ)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1226  Ⅎwnf 1325  ∃wex 1358  ∃!weu 1878  ∃*wmo 1879  {cab 2004  ∀wral 2280  ∃wrex 2281  {crab 2284

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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-coll 3842  ax-sep 3845

This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rab 2289  df-v 2533  df-in 2897  df-ss 2904

This theorem is referenced by: (None)