Theorem ifbi 3630

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 801	. . 3 |- ( ( ph <-> ps ) -> ( ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) <-> ( ( x e. A /\ ps ) \/ ( x e. B /\ -. ps ) ) ) )
5	4	abbidv 2350	. 2 |- ( ( ph <-> ps ) -> { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) } = { x | ( ( x e. A /\ ps ) \/ ( x e. B /\ -. ps ) ) } )
6		df-if 3608	. 2 |- if ( ph , A , B ) = { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) }
7		df-if 3608	. 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 716   = wceq 1398   e. wcel 2202   {cab 2217   ifcif 3607

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-if 3608

This theorem is referenced by:  ifbid  3631  ifbieq2i  3633  ifnebibdc  3655  fodjuomni  7391  fodjumkv  7402  nninfwlpoimlemg  7417  1tonninf  10749  lgsdi  15839  nninfsellemqall  16724  nninfomni  16728