Theorem elif 3614

Description: Membership in a conditional operator. (Contributed by NM, 14-Feb-2005.)

Assertion

Ref	Expression
elif	|- ( A e. if ( ph , B , C ) <-> ( ( ph /\ A e. B ) \/ ( -. ph /\ A e. C ) ) )

Proof of Theorem elif

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2811	. 2 |- ( A e. if ( ph , B , C ) -> A e. _V )
2		elex 2811	. . . 4 |- ( A e. B -> A e. _V )
3	2	adantl 277	. . 3 |- ( ( ph /\ A e. B ) -> A e. _V )
4		elex 2811	. . . 4 |- ( A e. C -> A e. _V )
5	4	adantl 277	. . 3 |- ( ( -. ph /\ A e. C ) -> A e. _V )
6	3, 5	jaoi 721	. 2 |- ( ( ( ph /\ A e. B ) \/ ( -. ph /\ A e. C ) ) -> A e. _V )
7		eleq1 2292	. . . . . 6 |- ( x = A -> ( x e. B <-> A e. B ) )
8	7	anbi1d 465	. . . . 5 |- ( x = A -> ( ( x e. B /\ ph ) <-> ( A e. B /\ ph ) ) )
9		eleq1 2292	. . . . . 6 |- ( x = A -> ( x e. C <-> A e. C ) )
10	9	anbi1d 465	. . . . 5 |- ( x = A -> ( ( x e. C /\ -. ph ) <-> ( A e. C /\ -. ph ) ) )
11	8, 10	orbi12d 798	. . . 4 |- ( x = A -> ( ( ( x e. B /\ ph ) \/ ( x e. C /\ -. ph ) ) <-> ( ( A e. B /\ ph ) \/ ( A e. C /\ -. ph ) ) ) )
12		df-if 3603	. . . 4 |- if ( ph , B , C ) = { x | ( ( x e. B /\ ph ) \/ ( x e. C /\ -. ph ) ) }
13	11, 12	elab2g 2950	. . 3 |- ( A e. _V -> ( A e. if ( ph , B , C ) <-> ( ( A e. B /\ ph ) \/ ( A e. C /\ -. ph ) ) ) )
14		ancom 266	. . . 4 |- ( ( A e. B /\ ph ) <-> ( ph /\ A e. B ) )
15		ancom 266	. . . 4 |- ( ( A e. C /\ -. ph ) <-> ( -. ph /\ A e. C ) )
16	14, 15	orbi12i 769	. . 3 |- ( ( ( A e. B /\ ph ) \/ ( A e. C /\ -. ph ) ) <-> ( ( ph /\ A e. B ) \/ ( -. ph /\ A e. C ) ) )
17	13, 16	bitrdi 196	. 2 |- ( A e. _V -> ( A e. if ( ph , B , C ) <-> ( ( ph /\ A e. B ) \/ ( -. ph /\ A e. C ) ) ) )
18	1, 6, 17	pm5.21nii 709	1 |- ( A e. if ( ph , B , C ) <-> ( ( ph /\ A e. B ) \/ ( -. ph /\ A e. C ) ) )

Syntax hints:   -. wn 3   /\ wa 104   <-> wb 105   \/ wo 713   = wceq 1395   e. wcel 2200   _Vcvv 2799   ifcif 3602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-if 3603

This theorem is referenced by:  iftrueb01  7404  pw1if  7406