Theorem ibllem

Description: Conditioned equality theorem for the if statement. (Contributed by Mario Carneiro, 31-Jul-2014)

	Ref	Expression
Hypothesis	ibllem.1	|- ( ( ph /\ x e. A ) -> B = C )
Assertion	ibllem	|- ( ph -> if ( ( x e. A /\ 0 <_ B ) , B , 0 ) = if ( ( x e. A /\ 0 <_ C ) , C , 0 ) )

Proof

Step	Hyp	Ref	Expression
1		ibllem.1	|- ( ( ph /\ x e. A ) -> B = C )
2	1	breq2d	|- ( ( ph /\ x e. A ) -> ( 0 <_ B <-> 0 <_ C ) )
3	2	pm5.32da	|- ( ph -> ( ( x e. A /\ 0 <_ B ) <-> ( x e. A /\ 0 <_ C ) ) )
4	3	ifbid	|- ( ph -> if ( ( x e. A /\ 0 <_ B ) , B , 0 ) = if ( ( x e. A /\ 0 <_ C ) , B , 0 ) )
5	1	adantrr	|- ( ( ph /\ ( x e. A /\ 0 <_ C ) ) -> B = C )
6	5	ifeq1da	|- ( ph -> if ( ( x e. A /\ 0 <_ C ) , B , 0 ) = if ( ( x e. A /\ 0 <_ C ) , C , 0 ) )
7	4 6	eqtrd	|- ( ph -> if ( ( x e. A /\ 0 <_ B ) , B , 0 ) = if ( ( x e. A /\ 0 <_ C ) , C , 0 ) )