Theorem repizf2 3936

Description: Replacement. This version of replacement is stronger than repizf 3894 in the sense that ph 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 3894 with ax-sep 3896. Another variation would be A. x e. w E* y ph -> { y | E. x ( x e. w /\ ph ) } e. _V but we don't have a proof of that yet. (Contributed by Jim Kingdon, 7-Sep-2018.)

Hypothesis

Ref	Expression
repizf2.1	|- F/ z ph

Assertion

Ref	Expression
repizf2	|- ( A. x e. w E* y ph -> E. z A. x e. { x e. w | E. y ph } E. y e. z ph )

Distinct variable group:   x, y, z, w

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

Proof of Theorem repizf2

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2604	. . 3 |- w e. _V
2	1	rabex 3922	. 2 |- { x e. w | E. y ph } e. _V
3		repizf2lem 3935	. . . 4 |- ( A. x e. w E* y ph <-> A. x e. { x e. w | E. y ph } E! y ph )
4		nfcv 2219	. . . . . 6 |- F/_ x v
5		nfrab1 2533	. . . . . 6 |- F/_ x { x e. w | E. y ph }
6	4, 5	raleqf 2545	. . . . 5 |- ( v = { x e. w | E. y ph } -> ( A. x e. v E! y ph <-> A. x e. { x e. w | E. y ph } E! y ph ) )
7		repizf2.1	. . . . . 6 |- F/ z ph
8	7	repizf 3894	. . . . 5 |- ( A. x e. v E! y ph -> E. z A. x e. v E. y e. z ph )
9	6, 8	syl6bir 162	. . . 4 |- ( v = { x e. w | E. y ph } -> ( A. x e. { x e. w | E. y ph } E! y ph -> E. z A. x e. v E. y e. z ph ) )
10	3, 9	syl5bi 150	. . 3 |- ( v = { x e. w | E. y ph } -> ( A. x e. w E* y ph -> E. z A. x e. v E. y e. z ph ) )
11		df-rab 2357	. . . . . 6 |- { x e. w | E. y ph } = { x | ( x e. w /\ E. y ph ) }
12		nfv 1461	. . . . . . . 8 |- F/ z x e. w
13	7	nfex 1568	. . . . . . . 8 |- F/ z E. y ph
14	12, 13	nfan 1497	. . . . . . 7 |- F/ z ( x e. w /\ E. y ph )
15	14	nfab 2223	. . . . . 6 |- F/_ z { x | ( x e. w /\ E. y ph ) }
16	11, 15	nfcxfr 2216	. . . . 5 |- F/_ z { x e. w | E. y ph }
17	16	nfeq2 2230	. . . 4 |- F/ z v = { x e. w | E. y ph }
18	4, 5	raleqf 2545	. . . 4 |- ( v = { x e. w | E. y ph } -> ( A. x e. v E. y e. z ph <-> A. x e. { x e. w | E. y ph } E. y e. z ph ) )
19	17, 18	exbid 1547	. . 3 |- ( v = { x e. w | E. y ph } -> ( E. z A. x e. v E. y e. z ph <-> E. z A. x e. { x e. w | E. y ph } E. y e. z ph ) )
20	10, 19	sylibd 147	. 2 |- ( v = { x e. w | E. y ph } -> ( A. x e. w E* y ph -> E. z A. x e. { x e. w | E. y ph } E. y e. z ph ) )
21	2, 20	vtocle 2672	1 |- ( A. x e. w E* y ph -> E. z A. x e. { x e. w | E. y ph } E. y e. z ph )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   F/wnf 1389   E.wex 1421   E!weu 1941   E*wmo 1942   {cab 2067   A.wral 2348   E.wrex 2349   {crab 2352

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rab 2357  df-v 2603  df-in 2979  df-ss 2986

This theorem is referenced by: (None)