Theorem ifbi 3626

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 672	. . . . 5 |- ( ( ph <-> ps ) -> ( -. ph <-> -. ps ) )
3	2	anbi2d 464	. . . 4 |- ( ( ph <-> ps ) -> ( ( x e. B /\ -. ph ) <-> ( x e. B /\ -. ps ) ) )
4	1, 3	orbi12d 800	. . 3 |- ( ( ph <-> ps ) -> ( ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) <-> ( ( x e. A /\ ps ) \/ ( x e. B /\ -. ps ) ) ) )
5	4	abbidv 2349	. 2 |- ( ( ph <-> ps ) -> { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) } = { x | ( ( x e. A /\ ps ) \/ ( x e. B /\ -. ps ) ) } )
6		df-if 3606	. 2 |- if ( ph , A , B ) = { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) }
7		df-if 3606	. 2 |- if ( ps , A , B ) = { x | ( ( x e. A /\ ps ) \/ ( x e. B /\ -. ps ) ) }
8	5, 6, 7	3eqtr4g 2289	1 |- ( ( ph <-> ps ) -> if ( ph , A , B ) = if ( ps , A , B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 715   = wceq 1397   e. wcel 2202   {cab 2217   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-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  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-if 3606

This theorem is referenced by:  ifbid  3627  ifbieq2i  3629  ifnebibdc  3651  fodjuomni  7347  fodjumkv  7358  nninfwlpoimlemg  7373  1tonninf  10702  lgsdi  15765  nninfsellemqall  16617  nninfomni  16621