Theorem repizf2 4222

Description: Replacement. This version of replacement is stronger than repizf 4176 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 4176 with ax-sep 4178. 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 2779	. . 3 |- w e. _V
2	1	rabex 4204	. 2 |- { x e. w | E. y ph } e. _V
3		repizf2lem 4221	. . . 4 |- ( A. x e. w E* y ph <-> A. x e. { x e. w | E. y ph } E! y ph )
4		nfcv 2350	. . . . . 6 |- F/_ x v
5		nfrab1 2688	. . . . . 6 |- F/_ x { x e. w | E. y ph }
6	4, 5	raleqf 2701	. . . . 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 4176	. . . . 5 |- ( A. x e. v E! y ph -> E. z A. x e. v E. y e. z ph )
9	6, 8	biimtrrdi 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 2495	. . . . . 6 |- { x e. w | E. y ph } = { x | ( x e. w /\ E. y ph ) }
12		nfv 1552	. . . . . . . 8 |- F/ z x e. w
13	7	nfex 1661	. . . . . . . 8 |- F/ z E. y ph
14	12, 13	nfan 1589	. . . . . . 7 |- F/ z ( x e. w /\ E. y ph )
15	14	nfab 2355	. . . . . 6 |- F/_ z { x | ( x e. w /\ E. y ph ) }
16	11, 15	nfcxfr 2347	. . . . 5 |- F/_ z { x e. w | E. y ph }
17	16	nfeq2 2362	. . . 4 |- F/ z v = { x e. w | E. y ph }
18	4, 5	raleqf 2701	. . . 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 1640	. . 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 2854	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 1373   F/wnf 1484   E.wex 1516   E!weu 2055   E*wmo 2056   {cab 2193   A.wral 2486   E.wrex 2487   {crab 2490

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189  ax-coll 4175  ax-sep 4178

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rab 2495  df-v 2778  df-in 3180  df-ss 3187

This theorem is referenced by: (None)