Theorem elif 3617

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 2814	. 2 |- ( A e. if ( ph , B , C ) -> A e. _V )
2		elex 2814	. . . 4 |- ( A e. B -> A e. _V )
3	2	adantl 277	. . 3 |- ( ( ph /\ A e. B ) -> A e. _V )
4		elex 2814	. . . 4 |- ( A e. C -> A e. _V )
5	4	adantl 277	. . 3 |- ( ( -. ph /\ A e. C ) -> A e. _V )
6	3, 5	jaoi 723	. 2 |- ( ( ( ph /\ A e. B ) \/ ( -. ph /\ A e. C ) ) -> A e. _V )
7		eleq1 2294	. . . . . 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 2294	. . . . . 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 800	. . . 4 |- ( x = A -> ( ( ( x e. B /\ ph ) \/ ( x e. C /\ -. ph ) ) <-> ( ( A e. B /\ ph ) \/ ( A e. C /\ -. ph ) ) ) )
12		df-if 3606	. . . 4 |- if ( ph , B , C ) = { x | ( ( x e. B /\ ph ) \/ ( x e. C /\ -. ph ) ) }
13	11, 12	elab2g 2953	. . 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 771	. . 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 711	1 |- ( A e. if ( ph , B , C ) <-> ( ( ph /\ A e. B ) \/ ( -. ph /\ A e. C ) ) )

Syntax hints:   -. wn 3   /\ wa 104   <-> wb 105   \/ wo 715   = wceq 1397   e. wcel 2202   _Vcvv 2802   ifcif 3605

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-if 3606

This theorem is referenced by:  rabsnif  3738  iftrueb01  7440  pw1if  7442  eupth2lem1  16308