Theorem ifbi 3415

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 456	. . . 4 |- ( ( ph <-> ps ) -> ( ( x e. A /\ ph ) <-> ( x e. A /\ ps ) ) )
2		id 19	. . . . . 6 |- ( ( ph <-> ps ) -> ( ph <-> ps ) )
3	2	notbid 628	. . . . 5 |- ( ( ph <-> ps ) -> ( -. ph <-> -. ps ) )
4	3	anbi2d 453	. . . 4 |- ( ( ph <-> ps ) -> ( ( x e. B /\ -. ph ) <-> ( x e. B /\ -. ps ) ) )
5	1, 4	orbi12d 743	. . 3 |- ( ( ph <-> ps ) -> ( ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) <-> ( ( x e. A /\ ps ) \/ ( x e. B /\ -. ps ) ) ) )
6	5	abbidv 2206	. 2 |- ( ( ph <-> ps ) -> { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) } = { x | ( ( x e. A /\ ps ) \/ ( x e. B /\ -. ps ) ) } )
7		df-if 3398	. 2 |- if ( ph , A , B ) = { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) }
8		df-if 3398	. 2 |- if ( ps , A , B ) = { x | ( ( x e. A /\ ps ) \/ ( x e. B /\ -. ps ) ) }
9	6, 7, 8	3eqtr4g 2146	1 |- ( ( ph <-> ps ) -> if ( ph , A , B ) = if ( ps , A , B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 665   = wceq 1290   e. wcel 1439   {cab 2075   ifcif 3397

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-11 1443  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-if 3398

This theorem is referenced by:  ifbid  3416  ifbieq2i  3418  fodjuomni  6865  1tonninf  9907  nninfsellemqall  12179  nninfomni  12183