Theorem reu8nf 3516

Description: Restricted uniqueness using implicit substitution. This version of reu8 3402 uses a non-freeness 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, y, w   ps, w   ch, y

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

Proof of Theorem reu8nf

Step	Hyp	Ref	Expression
1		nfv 1843	. . 3 |- F/ w ph
2		reu8nf.2	. . 3 |- F/ x ch
3		reu8nf.3	. . 3 |- ( x = w -> ( ph <-> ch ) )
4	1, 2, 3	cbvreu 3169	. 2 |- ( E! x e. A ph <-> E! w e. A ch )
5		reu8nf.4	. . 3 |- ( w = y -> ( ch <-> ps ) )
6	5	reu8 3402	. 2 |- ( E! w e. A ch <-> E. w e. A ( ch /\ A. y e. A ( ps -> w = y ) ) )
7		nfcv 2764	. . . . 5 |- F/_ x A
8		reu8nf.1	. . . . . 6 |- F/ x ps
9		nfv 1843	. . . . . 6 |- F/ x w = y
10	8, 9	nfim 1825	. . . . 5 |- F/ x ( ps -> w = y )
11	7, 10	nfral 2945	. . . 4 |- F/ x A. y e. A ( ps -> w = y )
12	2, 11	nfan 1828	. . 3 |- F/ x ( ch /\ A. y e. A ( ps -> w = y ) )
13		nfv 1843	. . 3 |- F/ w ( ph /\ A. y e. A ( ps -> x = y ) )
14	3	bicomd 213	. . . . 5 |- ( x = w -> ( ch <-> ph ) )
15	14	equcoms 1947	. . . 4 |- ( w = x -> ( ch <-> ph ) )
16		equequ1 1952	. . . . . 6 |- ( w = x -> ( w = y <-> x = y ) )
17	16	imbi2d 330	. . . . 5 |- ( w = x -> ( ( ps -> w = y ) <-> ( ps -> x = y ) ) )
18	17	ralbidv 2986	. . . 4 |- ( w = x -> ( A. y e. A ( ps -> w = y ) <-> A. y e. A ( ps -> x = y ) ) )
19	15, 18	anbi12d 747	. . 3 |- ( w = x -> ( ( ch /\ A. y e. A ( ps -> w = y ) ) <-> ( ph /\ A. y e. A ( ps -> x = y ) ) ) )
20	12, 13, 19	cbvrex 3168	. 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 286	1 |- ( E! x e. A ph <-> E. x e. A ( ph /\ A. y e. A ( ps -> x = y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   F/wnf 1708   A.wral 2912   E.wrex 2913   E!wreu 2914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-reu 2919

This theorem is referenced by:  reuccats1  13480