Theorem rabsnif 4101

Description: A restricted class abstraction restricted to a singleton is either the empty set or the singleton itself. (Contributed by AV, 12-Apr-2019.) (Proof shortened by AV, 21-Jul-2019.)

Hypothesis

Ref	Expression
rabsnif.f	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
rabsnif	|- { x e. { A } | ph } = if ( ps , { A } , (/) )

Distinct variable groups:   x, A   ps, x

Allowed substitution hint:   ph( x)

Proof of Theorem rabsnif

Step	Hyp	Ref	Expression
1		rabsnifsb 4100	. . 3 |- { x e. { A } | ph } = if ( [. A / x ]. ph , { A } , (/) )
2		rabsnif.f	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
3	2	sbcieg 3369	. . . 4 |- ( A e. _V -> ( [. A / x ]. ph <-> ps ) )
4	3	ifbid 3966	. . 3 |- ( A e. _V -> if ( [. A / x ]. ph , { A } , (/) ) = if ( ps , { A } , (/) ) )
5	1, 4	syl5eq 2520	. 2 |- ( A e. _V -> { x e. { A } | ph } = if ( ps , { A } , (/) ) )
6		rab0 3811	. . . 4 |- { x e. (/) | ph } = (/)
7		ifid 3981	. . . 4 |- if ( ps , (/) , (/) ) = (/)
8	6, 7	eqtr4i 2499	. . 3 |- { x e. (/) | ph } = if ( ps , (/) , (/) )
9		snprc 4096	. . . . 5 |- ( -. A e. _V <-> { A } = (/) )
10	9	biimpi 194	. . . 4 |- ( -. A e. _V -> { A } = (/) )
11		rabeq 3112	. . . 4 |- ( { A } = (/) -> { x e. { A } | ph } = { x e. (/) | ph } )
12	10, 11	syl 16	. . 3 |- ( -. A e. _V -> { x e. { A } | ph } = { x e. (/) | ph } )
13	10	ifeq1d 3962	. . 3 |- ( -. A e. _V -> if ( ps , { A } , (/) ) = if ( ps , (/) , (/) ) )
14	8, 12, 13	3eqtr4a 2534	. 2 |- ( -. A e. _V -> { x e. { A } | ph } = if ( ps , { A } , (/) ) )
15	5, 14	pm2.61i 164	1 |- { x e. { A } | ph } = if ( ps , { A } , (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   = wceq 1379   e. wcel 1767   {crab 2821   _Vcvv 3118   [.wsbc 3336   (/)c0 3790   ifcif 3944   {csn 4032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-nul 3791  df-if 3945  df-sn 4033

This theorem is referenced by:  suppsnop  6925  m1detdiag  18945