Theorem reu8nf 3113

Description: Restricted uniqueness using implicit substitution. This version of reu8 3002 uses a nonfreeness hypothesis for x and ps instead of distinct variable conditions. (Contributed by AV, 21-Jan-2022.)

Hypotheses

Ref	Expression
reu8nf.1	|- F/ x ps
reu8nf.2	|- F/ x ch
reu8nf.3	|- ( x = w -> ( ph <-> ch ) )
reu8nf.4	|- ( w = y -> ( ch <-> ps ) )

Assertion

Ref	Expression
reu8nf	|- ( E! x e. A ph <-> E. x e. A ( ph /\ A. y e. A ( ps -> x = y ) ) )

Distinct variable groups:   x, w, y, A   ph, w   ps, w   ch, y

Allowed substitution hints:   ph( x, y)   ps( x, y)   ch( x, w)

Proof of Theorem reu8nf

Step	Hyp	Ref	Expression
1		nfv 1576	. . 3 |- F/ w ph
2		reu8nf.2	. . 3 |- F/ x ch
3		reu8nf.3	. . 3 |- ( x = w -> ( ph <-> ch ) )
4	1, 2, 3	cbvreuw 2762	. 2 |- ( E! x e. A ph <-> E! w e. A ch )
5		reu8nf.4	. . 3 |- ( w = y -> ( ch <-> ps ) )
6	5	reu8 3002	. 2 |- ( E! w e. A ch <-> E. w e. A ( ch /\ A. y e. A ( ps -> w = y ) ) )
7		nfcv 2374	. . . . 5 |- F/_ x A
8		reu8nf.1	. . . . . 6 |- F/ x ps
9		nfv 1576	. . . . . 6 |- F/ x w = y
10	8, 9	nfim 1620	. . . . 5 |- F/ x ( ps -> w = y )
11	7, 10	nfralw 2569	. . . 4 |- F/ x A. y e. A ( ps -> w = y )
12	2, 11	nfan 1613	. . 3 |- F/ x ( ch /\ A. y e. A ( ps -> w = y ) )
13		nfv 1576	. . 3 |- F/ w ( ph /\ A. y e. A ( ps -> x = y ) )
14	3	bicomd 141	. . . . 5 |- ( x = w -> ( ch <-> ph ) )
15	14	equcoms 1756	. . . 4 |- ( w = x -> ( ch <-> ph ) )
16		equequ1 1760	. . . . . 6 |- ( w = x -> ( w = y <-> x = y ) )
17	16	imbi2d 230	. . . . 5 |- ( w = x -> ( ( ps -> w = y ) <-> ( ps -> x = y ) ) )
18	17	ralbidv 2532	. . . 4 |- ( w = x -> ( A. y e. A ( ps -> w = y ) <-> A. y e. A ( ps -> x = y ) ) )
19	15, 18	anbi12d 473	. . 3 |- ( w = x -> ( ( ch /\ A. y e. A ( ps -> w = y ) ) <-> ( ph /\ A. y e. A ( ps -> x = y ) ) ) )
20	12, 13, 19	cbvrexw 2761	. 2 |- ( E. w e. A ( ch /\ A. y e. A ( ps -> w = y ) ) <-> E. x e. A ( ph /\ A. y e. A ( ps -> x = y ) ) )
21	4, 6, 20	3bitri 206	1 |- ( E! x e. A ph <-> E. x e. A ( ph /\ A. y e. A ( ps -> x = y ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   F/wnf 1508   A.wral 2510   E.wrex 2511   E!wreu 2512

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-v 2804

This theorem is referenced by:  reuccatpfxs1  11327