Theorem ifbi 3623

Description: Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)

Assertion

Ref	Expression
ifbi	|- ( ( ph <-> ps ) -> if ( ph , A , B ) = if ( ps , A , B ) )

Proof of Theorem ifbi

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		anbi2 467	. . . 4 |- ( ( ph <-> ps ) -> ( ( x e. A /\ ph ) <-> ( x e. A /\ ps ) ) )
2		notbi 670	. . . . 5 |- ( ( ph <-> ps ) -> ( -. ph <-> -. ps ) )
3	2	anbi2d 464	. . . 4 |- ( ( ph <-> ps ) -> ( ( x e. B /\ -. ph ) <-> ( x e. B /\ -. ps ) ) )
4	1, 3	orbi12d 798	. . 3 |- ( ( ph <-> ps ) -> ( ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) <-> ( ( x e. A /\ ps ) \/ ( x e. B /\ -. ps ) ) ) )
5	4	abbidv 2347	. 2 |- ( ( ph <-> ps ) -> { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) } = { x | ( ( x e. A /\ ps ) \/ ( x e. B /\ -. ps ) ) } )
6		df-if 3603	. 2 |- if ( ph , A , B ) = { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) }
7		df-if 3603	. 2 |- if ( ps , A , B ) = { x | ( ( x e. A /\ ps ) \/ ( x e. B /\ -. ps ) ) }
8	5, 6, 7	3eqtr4g 2287	1 |- ( ( ph <-> ps ) -> if ( ph , A , B ) = if ( ps , A , B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   = wceq 1395   e. wcel 2200   {cab 2215   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-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  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-if 3603

This theorem is referenced by:  ifbid  3624  ifbieq2i  3626  ifnebibdc  3648  fodjuomni  7304  fodjumkv  7315  nninfwlpoimlemg  7330  1tonninf  10650  lgsdi  15701  nninfsellemqall  16312  nninfomni  16316