Theorem strfvnd

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

	Ref	Expression
Hypotheses	strfvnd.c	$$⊢E=\mathrm{Slot}N$$
	strfvnd.f	$$⊢\phi \to S\in V$$
Assertion	strfvnd	$$⊢\phi \to E\left(S\right)=S\left(N\right)$$

Proof

Step	Hyp	Ref	Expression
1		strfvnd.c	$$⊢E=\mathrm{Slot}N$$
2		strfvnd.f	$$⊢\phi \to S\in V$$
3		elex	$$⊢S\in V\to S\in \mathrm{V}$$
4		fveq1	$$⊢x=S\to x\left(N\right)=S\left(N\right)$$
5		df-slot	$$⊢\mathrm{Slot}N=\left(x\in \mathrm{V}⟼x\left(N\right)\right)$$
6	1 5	eqtri	$$⊢E=\left(x\in \mathrm{V}⟼x\left(N\right)\right)$$
7		fvex	$$⊢S\left(N\right)\in \mathrm{V}$$
8	4 6 7	fvmpt	$$⊢S\in \mathrm{V}\to E\left(S\right)=S\left(N\right)$$
9	2 3 8	3syl	$$⊢\phi \to E\left(S\right)=S\left(N\right)$$