Theorem repizf2 4164

Description: Replacement. This version of replacement is stronger than repizf 4121 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 4121 with ax-sep 4123. 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 2742	. . 3 |- w e. _V
2	1	rabex 4149	. 2 |- { x e. w | E. y ph } e. _V
3		repizf2lem 4163	. . . 4 |- ( A. x e. w E* y ph <-> A. x e. { x e. w | E. y ph } E! y ph )
4		nfcv 2319	. . . . . 6 |- F/_ x v
5		nfrab1 2657	. . . . . 6 |- F/_ x { x e. w | E. y ph }
6	4, 5	raleqf 2669	. . . . 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 4121	. . . . 5 |- ( A. x e. v E! y ph -> E. z A. x e. v E. y e. z ph )
9	6, 8	syl6bir 164	. . . 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	biimtrid 152	. . 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 2464	. . . . . 6 |- { x e. w | E. y ph } = { x | ( x e. w /\ E. y ph ) }
12		nfv 1528	. . . . . . . 8 |- F/ z x e. w
13	7	nfex 1637	. . . . . . . 8 |- F/ z E. y ph
14	12, 13	nfan 1565	. . . . . . 7 |- F/ z ( x e. w /\ E. y ph )
15	14	nfab 2324	. . . . . 6 |- F/_ z { x | ( x e. w /\ E. y ph ) }
16	11, 15	nfcxfr 2316	. . . . 5 |- F/_ z { x e. w | E. y ph }
17	16	nfeq2 2331	. . . 4 |- F/ z v = { x e. w | E. y ph }
18	4, 5	raleqf 2669	. . . 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 1616	. . 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 149	. 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 2813	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 104   = wceq 1353   F/wnf 1460   E.wex 1492   E!weu 2026   E*wmo 2027   {cab 2163   A.wral 2455   E.wrex 2456   {crab 2459

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-coll 4120  ax-sep 4123

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rab 2464  df-v 2741  df-in 3137  df-ss 3144

This theorem is referenced by: (None)