Theorem ifbi 3540

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 463	. . . 4 |- ( ( ph <-> ps ) -> ( ( x e. A /\ ph ) <-> ( x e. A /\ ps ) ) )
2		notbi 656	. . . . 5 |- ( ( ph <-> ps ) -> ( -. ph <-> -. ps ) )
3	2	anbi2d 460	. . . 4 |- ( ( ph <-> ps ) -> ( ( x e. B /\ -. ph ) <-> ( x e. B /\ -. ps ) ) )
4	1, 3	orbi12d 783	. . 3 |- ( ( ph <-> ps ) -> ( ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) <-> ( ( x e. A /\ ps ) \/ ( x e. B /\ -. ps ) ) ) )
5	4	abbidv 2284	. 2 |- ( ( ph <-> ps ) -> { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) } = { x | ( ( x e. A /\ ps ) \/ ( x e. B /\ -. ps ) ) } )
6		df-if 3521	. 2 |- if ( ph , A , B ) = { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) }
7		df-if 3521	. 2 |- if ( ps , A , B ) = { x | ( ( x e. A /\ ps ) \/ ( x e. B /\ -. ps ) ) }
8	5, 6, 7	3eqtr4g 2224	1 |- ( ( ph <-> ps ) -> if ( ph , A , B ) = if ( ps , A , B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   = wceq 1343   e. wcel 2136   {cab 2151   ifcif 3520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-if 3521

This theorem is referenced by:  ifbid  3541  ifbieq2i  3543  fodjuomni  7113  fodjumkv  7124  1tonninf  10375  lgsdi  13578  nninfsellemqall  13895  nninfomni  13899