Theorem rabsnif 4032

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 4031	. . 3 |- { x e. { A } | ph } = if ( [. A / x ]. ph , { A } , (/) )
2		rabsnif.f	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
3	2	sbcieg 3288	. . . 4 |- ( A e. _V -> ( [. A / x ]. ph <-> ps ) )
4	3	ifbid 3894	. . 3 |- ( A e. _V -> if ( [. A / x ]. ph , { A } , (/) ) = if ( ps , { A } , (/) ) )
5	1, 4	syl5eq 2517	. 2 |- ( A e. _V -> { x e. { A } | ph } = if ( ps , { A } , (/) ) )
6		rab0 3756	. . . 4 |- { x e. (/) | ph } = (/)
7		ifid 3909	. . . 4 |- if ( ps , (/) , (/) ) = (/)
8	6, 7	eqtr4i 2496	. . 3 |- { x e. (/) | ph } = if ( ps , (/) , (/) )
9		snprc 4027	. . . . 5 |- ( -. A e. _V <-> { A } = (/) )
10	9	biimpi 199	. . . 4 |- ( -. A e. _V -> { A } = (/) )
11		rabeq 3024	. . . 4 |- ( { A } = (/) -> { x e. { A } | ph } = { x e. (/) | ph } )
12	10, 11	syl 17	. . 3 |- ( -. A e. _V -> { x e. { A } | ph } = { x e. (/) | ph } )
13	10	ifeq1d 3890	. . 3 |- ( -. A e. _V -> if ( ps , { A } , (/) ) = if ( ps , (/) , (/) ) )
14	8, 12, 13	3eqtr4a 2531	. 2 |- ( -. A e. _V -> { x e. { A } | ph } = if ( ps , { A } , (/) ) )
15	5, 14	pm2.61i 169	1 |- { x e. { A } | ph } = if ( ps , { A } , (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   = wceq 1452   e. wcel 1904   {crab 2760   _Vcvv 3031   [.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-3an 1009  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-ne 2643  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-nul 3723  df-if 3873  df-sn 3960

This theorem is referenced by:  suppsnop  6947  m1detdiag  19699  1loopgrvd2  39725  1hevtxdg1  39728  1egrvtxdg1  39732