Theorem repizf2 4141

Description: Replacement. This version of replacement is stronger than repizf 4098 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 4098 with ax-sep 4100. 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 2729	. . 3 |- w e. _V
2	1	rabex 4126	. 2 |- { x e. w | E. y ph } e. _V
3		repizf2lem 4140	. . . 4 |- ( A. x e. w E* y ph <-> A. x e. { x e. w | E. y ph } E! y ph )
4		nfcv 2308	. . . . . 6 |- F/_ x v
5		nfrab1 2645	. . . . . 6 |- F/_ x { x e. w | E. y ph }
6	4, 5	raleqf 2657	. . . . 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 4098	. . . . 5 |- ( A. x e. v E! y ph -> E. z A. x e. v E. y e. z ph )
9	6, 8	syl6bir 163	. . . 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 151	. . 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 2453	. . . . . 6 |- { x e. w | E. y ph } = { x | ( x e. w /\ E. y ph ) }
12		nfv 1516	. . . . . . . 8 |- F/ z x e. w
13	7	nfex 1625	. . . . . . . 8 |- F/ z E. y ph
14	12, 13	nfan 1553	. . . . . . 7 |- F/ z ( x e. w /\ E. y ph )
15	14	nfab 2313	. . . . . 6 |- F/_ z { x | ( x e. w /\ E. y ph ) }
16	11, 15	nfcxfr 2305	. . . . 5 |- F/_ z { x e. w | E. y ph }
17	16	nfeq2 2320	. . . 4 |- F/ z v = { x e. w | E. y ph }
18	4, 5	raleqf 2657	. . . 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 1604	. . 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 148	. 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 2800	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 103   = wceq 1343   F/wnf 1448   E.wex 1480   E!weu 2014   E*wmo 2015   {cab 2151   A.wral 2444   E.wrex 2445   {crab 2448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-coll 4097  ax-sep 4100

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rab 2453  df-v 2728  df-in 3122  df-ss 3129

This theorem is referenced by: (None)