Theorem cdlemk40f

Description: TODO: fix comment. (Contributed by NM, 31-Jul-2013)

	Ref	Expression
Hypotheses	cdlemk40.x	|- X = ( iota_ z e. T ph )
	cdlemk40.u	|- U = ( g e. T |-> if ( F = N , g , X ) )
Assertion	cdlemk40f	|- ( ( F =/= N /\ G e. T ) -> ( U ` G ) = [_ G / g ]_ X )

Proof

Step	Hyp	Ref	Expression
1		cdlemk40.x	|- X = ( iota_ z e. T ph )
2		cdlemk40.u	|- U = ( g e. T |-> if ( F = N , g , X ) )
3	1 2	cdlemk40	|- ( G e. T -> ( U ` G ) = if ( F = N , G , [_ G / g ]_ X ) )
4		ifnefalse	|- ( F =/= N -> if ( F = N , G , [_ G / g ]_ X ) = [_ G / g ]_ X )
5	3 4	sylan9eqr	|- ( ( F =/= N /\ G e. T ) -> ( U ` G ) = [_ G / g ]_ X )