Theorem rabsnifsb 4031

Description: A restricted class abstraction restricted to a singleton is either the empty set or the singleton itself. (Contributed by AV, 21-Jul-2019.)

Assertion

Ref	Expression
rabsnifsb	|- { x e. { A } | ph } = if ( [. A / x ]. ph , { A } , (/) )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem rabsnifsb

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elsni 3985	. . . . . . . 8 |- ( x e. { A } -> x = A )
2		sbceq1a 3266	. . . . . . . . 9 |- ( x = A -> ( ph <-> [. A / x ]. ph ) )
3	2	biimpd 212	. . . . . . . 8 |- ( x = A -> ( ph -> [. A / x ]. ph ) )
4	1, 3	syl 17	. . . . . . 7 |- ( x e. { A } -> ( ph -> [. A / x ]. ph ) )
5	4	imdistani 704	. . . . . 6 |- ( ( x e. { A } /\ ph ) -> ( x e. { A } /\ [. A / x ]. ph ) )
6	5	orcd 399	. . . . 5 |- ( ( x e. { A } /\ ph ) -> ( ( x e. { A } /\ [. A / x ]. ph ) \/ ( x e. (/) /\ -. [. A / x ]. ph ) ) )
7	2	bicomd 206	. . . . . . . . 9 |- ( x = A -> ( [. A / x ]. ph <-> ph ) )
8	1, 7	syl 17	. . . . . . . 8 |- ( x e. { A } -> ( [. A / x ]. ph <-> ph ) )
9	8	biimpd 212	. . . . . . 7 |- ( x e. { A } -> ( [. A / x ]. ph -> ph ) )
10	9	imdistani 704	. . . . . 6 |- ( ( x e. { A } /\ [. A / x ]. ph ) -> ( x e. { A } /\ ph ) )
11		noel 3726	. . . . . . . 8 |- -. x e. (/)
12	11	pm2.21i 136	. . . . . . 7 |- ( x e. (/) -> ( x e. { A } /\ ph ) )
13	12	adantr 472	. . . . . 6 |- ( ( x e. (/) /\ -. [. A / x ]. ph ) -> ( x e. { A } /\ ph ) )
14	10, 13	jaoi 386	. . . . 5 |- ( ( ( x e. { A } /\ [. A / x ]. ph ) \/ ( x e. (/) /\ -. [. A / x ]. ph ) ) -> ( x e. { A } /\ ph ) )
15	6, 14	impbii 192	. . . 4 |- ( ( x e. { A } /\ ph ) <-> ( ( x e. { A } /\ [. A / x ]. ph ) \/ ( x e. (/) /\ -. [. A / x ]. ph ) ) )
16	15	abbii 2587	. . 3 |- { x | ( x e. { A } /\ ph ) } = { x | ( ( x e. { A } /\ [. A / x ]. ph ) \/ ( x e. (/) /\ -. [. A / x ]. ph ) ) }
17		nfv 1769	. . . 4 |- F/ y ( ( x e. { A } /\ [. A / x ]. ph ) \/ ( x e. (/) /\ -. [. A / x ]. ph ) )
18		nfv 1769	. . . . . 6 |- F/ x y e. { A }
19		nfsbc1v 3275	. . . . . 6 |- F/ x [. A / x ]. ph
20	18, 19	nfan 2031	. . . . 5 |- F/ x ( y e. { A } /\ [. A / x ]. ph )
21		nfv 1769	. . . . . 6 |- F/ x y e. (/)
22	19	nfn 2003	. . . . . 6 |- F/ x -. [. A / x ]. ph
23	21, 22	nfan 2031	. . . . 5 |- F/ x ( y e. (/) /\ -. [. A / x ]. ph )
24	20, 23	nfor 2038	. . . 4 |- F/ x ( ( y e. { A } /\ [. A / x ]. ph ) \/ ( y e. (/) /\ -. [. A / x ]. ph ) )
25		eleq1 2537	. . . . . 6 |- ( x = y -> ( x e. { A } <-> y e. { A } ) )
26	25	anbi1d 719	. . . . 5 |- ( x = y -> ( ( x e. { A } /\ [. A / x ]. ph ) <-> ( y e. { A } /\ [. A / x ]. ph ) ) )
27		eleq1 2537	. . . . . 6 |- ( x = y -> ( x e. (/) <-> y e. (/) ) )
28	27	anbi1d 719	. . . . 5 |- ( x = y -> ( ( x e. (/) /\ -. [. A / x ]. ph ) <-> ( y e. (/) /\ -. [. A / x ]. ph ) ) )
29	26, 28	orbi12d 724	. . . 4 |- ( x = y -> ( ( ( x e. { A } /\ [. A / x ]. ph ) \/ ( x e. (/) /\ -. [. A / x ]. ph ) ) <-> ( ( y e. { A } /\ [. A / x ]. ph ) \/ ( y e. (/) /\ -. [. A / x ]. ph ) ) ) )
30	17, 24, 29	cbvab 2594	. . 3 |- { x | ( ( x e. { A } /\ [. A / x ]. ph ) \/ ( x e. (/) /\ -. [. A / x ]. ph ) ) } = { y | ( ( y e. { A } /\ [. A / x ]. ph ) \/ ( y e. (/) /\ -. [. A / x ]. ph ) ) }
31	16, 30	eqtri 2493	. 2 |- { x | ( x e. { A } /\ ph ) } = { y | ( ( y e. { A } /\ [. A / x ]. ph ) \/ ( y e. (/) /\ -. [. A / x ]. ph ) ) }
32		df-rab 2765	. 2 |- { x e. { A } | ph } = { x | ( x e. { A } /\ ph ) }
33		df-if 3873	. 2 |- if ( [. A / x ]. ph , { A } , (/) ) = { y | ( ( y e. { A } /\ [. A / x ]. ph ) \/ ( y e. (/) /\ -. [. A / x ]. ph ) ) }
34	31, 32, 33	3eqtr4i 2503	1 |- { x e. { A } | ph } = if ( [. A / x ]. ph , { A } , (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   \/ wo 375   /\ wa 376   = wceq 1452   e. wcel 1904   {cab 2457   {crab 2760   [.wsbc 3255   (/)c0 3722   ifcif 3872   {csn 3959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-nul 3723  df-if 3873  df-sn 3960

This theorem is referenced by:  rabsnif  4032  rabrsn  4033