Theorem ifeq2

Description: Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004) (Revised by Mario Carneiro, 8-Sep-2013)

	Ref	Expression
Assertion	ifeq2	|- ( A = B -> if ( ph , C , A ) = if ( ph , C , B ) )

Proof

Step	Hyp	Ref	Expression
1		rabeq	|- ( A = B -> { x e. A | -. ph } = { x e. B | -. ph } )
2	1	uneq2d	|- ( A = B -> ( { x e. C | ph } u. { x e. A | -. ph } ) = ( { x e. C | ph } u. { x e. B | -. ph } ) )
3		dfif6	|- if ( ph , C , A ) = ( { x e. C | ph } u. { x e. A | -. ph } )
4		dfif6	|- if ( ph , C , B ) = ( { x e. C | ph } u. { x e. B | -. ph } )
5	2 3 4	3eqtr4g	|- ( A = B -> if ( ph , C , A ) = if ( ph , C , B ) )