Theorem rabsnif 4258

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 4257	. . 3 |- { x e. { A } | ph } = if ( [. A / x ]. ph , { A } , (/) )
2		rabsnif.f	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
3	2	sbcieg 3468	. . . 4 |- ( A e. _V -> ( [. A / x ]. ph <-> ps ) )
4	3	ifbid 4108	. . 3 |- ( A e. _V -> if ( [. A / x ]. ph , { A } , (/) ) = if ( ps , { A } , (/) ) )
5	1, 4	syl5eq 2668	. 2 |- ( A e. _V -> { x e. { A } | ph } = if ( ps , { A } , (/) ) )
6		rab0 3955	. . . 4 |- { x e. (/) | ph } = (/)
7		ifid 4125	. . . 4 |- if ( ps , (/) , (/) ) = (/)
8	6, 7	eqtr4i 2647	. . 3 |- { x e. (/) | ph } = if ( ps , (/) , (/) )
9		snprc 4253	. . . . 5 |- ( -. A e. _V <-> { A } = (/) )
10	9	biimpi 206	. . . 4 |- ( -. A e. _V -> { A } = (/) )
11	10	rabeqdv 3194	. . 3 |- ( -. A e. _V -> { x e. { A } | ph } = { x e. (/) | ph } )
12	10	ifeq1d 4104	. . 3 |- ( -. A e. _V -> if ( ps , { A } , (/) ) = if ( ps , (/) , (/) ) )
13	8, 11, 12	3eqtr4a 2682	. 2 |- ( -. A e. _V -> { x e. { A } | ph } = if ( ps , { A } , (/) ) )
14	5, 13	pm2.61i 176	1 |- { x e. { A } | ph } = if ( ps , { A } , (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   [.wsbc 3435   (/)c0 3915   ifcif 4086   {csn 4177

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-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-nul 3916  df-if 4087  df-sn 4178

This theorem is referenced by:  suppsnop  7309  m1detdiag  20403  1loopgrvd2  26399  1hevtxdg1  26402  1egrvtxdg1  26405