Theorem repizf2 4258

Description: Replacement. This version of replacement is stronger than repizf 4210 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 4210 with ax-sep 4212. 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 2806	. . 3 ⊢ 𝑤 ∈ V
2	1	rabex 4239	. 2 ⊢ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} ∈ V
3		repizf2lem 4257	. . . 4 ⊢ (∀𝑥 ∈ 𝑤 ∃*𝑦𝜑 ↔ ∀𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑)
4		nfcv 2375	. . . . . 6 ⊢ Ⅎ𝑥𝑣
5		nfrab1 2714	. . . . . 6 ⊢ Ⅎ𝑥{𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑}
6	4, 5	raleqf 2727	. . . . 5 ⊢ (𝑣 = {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} → (∀𝑥 ∈ 𝑣 ∃!𝑦𝜑 ↔ ∀𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑))
7		repizf2.1	. . . . . 6 ⊢ Ⅎ𝑧𝜑
8	7	repizf 4210	. . . . 5 ⊢ (∀𝑥 ∈ 𝑣 ∃!𝑦𝜑 → ∃𝑧∀𝑥 ∈ 𝑣 ∃𝑦 ∈ 𝑧 𝜑)
9	6, 8	biimtrrdi 164	. . . 4 ⊢ (𝑣 = {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} → (∀𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑 → ∃𝑧∀𝑥 ∈ 𝑣 ∃𝑦 ∈ 𝑧 𝜑))
10	3, 9	biimtrid 152	. . 3 ⊢ (𝑣 = {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} → (∀𝑥 ∈ 𝑤 ∃*𝑦𝜑 → ∃𝑧∀𝑥 ∈ 𝑣 ∃𝑦 ∈ 𝑧 𝜑))
11		df-rab 2520	. . . . . 6 ⊢ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} = {𝑥 ∣ (𝑥 ∈ 𝑤 ∧ ∃𝑦𝜑)}
12		nfv 1577	. . . . . . . 8 ⊢ Ⅎ𝑧 𝑥 ∈ 𝑤
13	7	nfex 1686	. . . . . . . 8 ⊢ Ⅎ𝑧∃𝑦𝜑
14	12, 13	nfan 1614	. . . . . . 7 ⊢ Ⅎ𝑧(𝑥 ∈ 𝑤 ∧ ∃𝑦𝜑)
15	14	nfab 2380	. . . . . 6 ⊢ Ⅎ𝑧{𝑥 ∣ (𝑥 ∈ 𝑤 ∧ ∃𝑦𝜑)}
16	11, 15	nfcxfr 2372	. . . . 5 ⊢ Ⅎ𝑧{𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑}
17	16	nfeq2 2387	. . . 4 ⊢ Ⅎ𝑧 𝑣 = {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑}
18	4, 5	raleqf 2727	. . . 4 ⊢ (𝑣 = {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} → (∀𝑥 ∈ 𝑣 ∃𝑦 ∈ 𝑧 𝜑 ↔ ∀𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑}∃𝑦 ∈ 𝑧 𝜑))
19	17, 18	exbid 1665	. . 3 ⊢ (𝑣 = {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} → (∃𝑧∀𝑥 ∈ 𝑣 ∃𝑦 ∈ 𝑧 𝜑 ↔ ∃𝑧∀𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑}∃𝑦 ∈ 𝑧 𝜑))
20	10, 19	sylibd 149	. 2 ⊢ (𝑣 = {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} → (∀𝑥 ∈ 𝑤 ∃*𝑦𝜑 → ∃𝑧∀𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑}∃𝑦 ∈ 𝑧 𝜑))
21	2, 20	vtocle 2881	1 ⊢ (∀𝑥 ∈ 𝑤 ∃*𝑦𝜑 → ∃𝑧∀𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑}∃𝑦 ∈ 𝑧 𝜑)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398  Ⅎwnf 1509  ∃wex 1541  ∃!weu 2079  ∃*wmo 2080  {cab 2217  ∀wral 2511  ∃wrex 2512  {crab 2515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213  ax-coll 4209  ax-sep 4212

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rab 2520  df-v 2805  df-in 3207  df-ss 3214

This theorem is referenced by: (None)