Theorem dfif6

Description: An alternate definition of the conditional operator df-if as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013)

	Ref	Expression
Assertion	dfif6	|- if ( ph , A , B ) = ( { x e. A | ph } u. { x e. B | -. ph } )

Proof

Step	Hyp	Ref	Expression
1		unab	|- ( { x | ( x e. A /\ ph ) } u. { x | ( x e. B /\ -. ph ) } ) = { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) }
2		df-rab	|- { x e. A | ph } = { x | ( x e. A /\ ph ) }
3		df-rab	|- { x e. B | -. ph } = { x | ( x e. B /\ -. ph ) }
4	2 3	uneq12i	|- ( { x e. A | ph } u. { x e. B | -. ph } ) = ( { x | ( x e. A /\ ph ) } u. { x | ( x e. B /\ -. ph ) } )
5		df-if	|- if ( ph , A , B ) = { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) }
6	1 4 5	3eqtr4ri	|- if ( ph , A , B ) = ( { x e. A | ph } u. { x e. B | -. ph } )