Theorem strfvnd

Description: Deduction version of strfvn . (Contributed by Mario Carneiro, 15-Nov-2014)

	Ref	Expression
Hypotheses	strfvnd.c	|- E = Slot N
	strfvnd.f	|- ( ph -> S e. V )
Assertion	strfvnd	|- ( ph -> ( E ` S ) = ( S ` N ) )

Proof

Step	Hyp	Ref	Expression
1		strfvnd.c	|- E = Slot N
2		strfvnd.f	|- ( ph -> S e. V )
3		elex	|- ( S e. V -> S e. _V )
4		fveq1	|- ( x = S -> ( x ` N ) = ( S ` N ) )
5		df-slot	|- Slot N = ( x e. _V |-> ( x ` N ) )
6	1 5	eqtri	|- E = ( x e. _V |-> ( x ` N ) )
7		fvex	|- ( S ` N ) e. _V
8	4 6 7	fvmpt	|- ( S e. _V -> ( E ` S ) = ( S ` N ) )
9	2 3 8	3syl	|- ( ph -> ( E ` S ) = ( S ` N ) )