Theorem repizf2 3915

Description: Replacement. This version of replacement is stronger than repizf 3873 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 3873 with ax-sep 3875. 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 2560	. . 3 |- w e. _V
2	1	rabex 3901	. 2 |- { x e. w | E. y ph } e. _V
3		repizf2lem 3914	. . . 4 |- ( A. x e. w E* y ph <-> A. x e. { x e. w | E. y ph } E! y ph )
4		nfcv 2178	. . . . . 6 |- F/_ x v
5		nfrab1 2489	. . . . . 6 |- F/_ x { x e. w | E. y ph }
6	4, 5	raleqf 2501	. . . . 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 3873	. . . . 5 |- ( A. x e. v E! y ph -> E. z A. x e. v E. y e. z ph )
9	6, 8	syl6bir 153	. . . 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 141	. . 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 2315	. . . . . 6 |- { x e. w | E. y ph } = { x | ( x e. w /\ E. y ph ) }
12		nfv 1421	. . . . . . . 8 |- F/ z x e. w
13	7	nfex 1528	. . . . . . . 8 |- F/ z E. y ph
14	12, 13	nfan 1457	. . . . . . 7 |- F/ z ( x e. w /\ E. y ph )
15	14	nfab 2182	. . . . . 6 |- F/_ z { x | ( x e. w /\ E. y ph ) }
16	11, 15	nfcxfr 2175	. . . . 5 |- F/_ z { x e. w | E. y ph }
17	16	nfeq2 2189	. . . 4 |- F/ z v = { x e. w | E. y ph }
18	4, 5	raleqf 2501	. . . 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 1507	. . 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 138	. 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 2627	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 97   = wceq 1243   F/wnf 1349   E.wex 1381   E!weu 1900   E*wmo 1901   {cab 2026   A.wral 2306   E.wrex 2307   {crab 2310

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rab 2315  df-v 2559  df-in 2924  df-ss 2931

This theorem is referenced by: (None)