Theorem reu8nf 3087

Description: Restricted uniqueness using implicit substitution. This version of reu8 2976 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 1552	. . 3 |- F/ w ph
2		reu8nf.2	. . 3 |- F/ x ch
3		reu8nf.3	. . 3 |- ( x = w -> ( ph <-> ch ) )
4	1, 2, 3	cbvreuw 2737	. 2 |- ( E! x e. A ph <-> E! w e. A ch )
5		reu8nf.4	. . 3 |- ( w = y -> ( ch <-> ps ) )
6	5	reu8 2976	. 2 |- ( E! w e. A ch <-> E. w e. A ( ch /\ A. y e. A ( ps -> w = y ) ) )
7		nfcv 2350	. . . . 5 |- F/_ x A
8		reu8nf.1	. . . . . 6 |- F/ x ps
9		nfv 1552	. . . . . 6 |- F/ x w = y
10	8, 9	nfim 1596	. . . . 5 |- F/ x ( ps -> w = y )
11	7, 10	nfralw 2545	. . . 4 |- F/ x A. y e. A ( ps -> w = y )
12	2, 11	nfan 1589	. . 3 |- F/ x ( ch /\ A. y e. A ( ps -> w = y ) )
13		nfv 1552	. . 3 |- F/ w ( ph /\ A. y e. A ( ps -> x = y ) )
14	3	bicomd 141	. . . . 5 |- ( x = w -> ( ch <-> ph ) )
15	14	equcoms 1732	. . . 4 |- ( w = x -> ( ch <-> ph ) )
16		equequ1 1736	. . . . . 6 |- ( w = x -> ( w = y <-> x = y ) )
17	16	imbi2d 230	. . . . 5 |- ( w = x -> ( ( ps -> w = y ) <-> ( ps -> x = y ) ) )
18	17	ralbidv 2508	. . . 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 2736	. 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 1484   A.wral 2486   E.wrex 2487   E!wreu 2488

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

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-rex 2492  df-reu 2493  df-rmo 2494  df-v 2778

This theorem is referenced by:  reuccatpfxs1  11238