Theorem rabsnif 4055

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 4054	. . 3 |- { x e. { A } | ph } = if ( [. A / x ]. ph , { A } , (/) )
2		rabsnif.f	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
3	2	sbcieg 3327	. . . 4 |- ( A e. _V -> ( [. A / x ]. ph <-> ps ) )
4	3	ifbid 3922	. . 3 |- ( A e. _V -> if ( [. A / x ]. ph , { A } , (/) ) = if ( ps , { A } , (/) ) )
5	1, 4	syl5eq 2507	. 2 |- ( A e. _V -> { x e. { A } | ph } = if ( ps , { A } , (/) ) )
6		rab0 3769	. . . 4 |- { x e. (/) | ph } = (/)
7		ifid 3937	. . . 4 |- if ( ps , (/) , (/) ) = (/)
8	6, 7	eqtr4i 2486	. . 3 |- { x e. (/) | ph } = if ( ps , (/) , (/) )
9		snprc 4050	. . . . 5 |- ( -. A e. _V <-> { A } = (/) )
10	9	biimpi 194	. . . 4 |- ( -. A e. _V -> { A } = (/) )
11		rabeq 3072	. . . 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 3918	. . 3 |- ( -. A e. _V -> if ( ps , { A } , (/) ) = if ( ps , (/) , (/) ) )
14	8, 12, 13	3eqtr4a 2521	. 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 1370   e. wcel 1758   {crab 2803   _Vcvv 3078   [.wsbc 3294   (/)c0 3748   ifcif 3902   {csn 3988

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-nul 3749  df-if 3903  df-sn 3989

This theorem is referenced by:  suppsnop  6817  m1detdiag  18545