Theorem elif 3649

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 2833	. 2 |- ( A e. if ( ph , B , C ) -> A e. _V )
2		elex 2833	. . . 4 |- ( A e. B -> A e. _V )
3	2	adantl 277	. . 3 |- ( ( ph /\ A e. B ) -> A e. _V )
4		elex 2833	. . . 4 |- ( A e. C -> A e. _V )
5	4	adantl 277	. . 3 |- ( ( -. ph /\ A e. C ) -> A e. _V )
6	3, 5	jaoi 728	. 2 |- ( ( ( ph /\ A e. B ) \/ ( -. ph /\ A e. C ) ) -> A e. _V )
7		eleq1 2301	. . . . . 6 |- ( x = A -> ( x e. B <-> A e. B ) )
8	7	anbi1d 469	. . . . 5 |- ( x = A -> ( ( x e. B /\ ph ) <-> ( A e. B /\ ph ) ) )
9		eleq1 2301	. . . . . 6 |- ( x = A -> ( x e. C <-> A e. C ) )
10	9	anbi1d 469	. . . . 5 |- ( x = A -> ( ( x e. C /\ -. ph ) <-> ( A e. C /\ -. ph ) ) )
11	8, 10	orbi12d 805	. . . 4 |- ( x = A -> ( ( ( x e. B /\ ph ) \/ ( x e. C /\ -. ph ) ) <-> ( ( A e. B /\ ph ) \/ ( A e. C /\ -. ph ) ) ) )
12		df-if 3636	. . . 4 |- if ( ph , B , C ) = { x | ( ( x e. B /\ ph ) \/ ( x e. C /\ -. ph ) ) }
13	11, 12	elab2g 2973	. . 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 776	. . 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 716	1 |- ( A e. if ( ph , B , C ) <-> ( ( ph /\ A e. B ) \/ ( -. ph /\ A e. C ) ) )

Syntax hints:   -. wn 3   /\ wa 104   <-> wb 105   \/ wo 720   = wceq 1402   e. wcel 2209   _Vcvv 2821   ifcif 3635

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-if 3636

This theorem is referenced by:  rabsnif  3774  iftrueb01  7572  pw1if  7574  eupth2lem1  16613