Theorem fvmptd

Description: Deduction version of fvmpt . (Contributed by Scott Fenton, 18-Feb-2013) (Revised by Mario Carneiro, 31-Aug-2015) (Proof shortened by AV, 29-Mar-2024)

	Ref	Expression
Hypotheses	fvmptd.1	$$⊢\phi \to F=\left(x\in D⟼B\right)$$
	fvmptd.2	$$⊢\left(\phi \wedge x=A\right)\to B=C$$
	fvmptd.3	$$⊢\phi \to A\in D$$
	fvmptd.4	$$⊢\phi \to C\in V$$
Assertion	fvmptd	$$⊢\phi \to F\left(A\right)=C$$

Proof

Step	Hyp	Ref	Expression
1		fvmptd.1	$$⊢\phi \to F=\left(x\in D⟼B\right)$$
2		fvmptd.2	$$⊢\left(\phi \wedge x=A\right)\to B=C$$
3		fvmptd.3	$$⊢\phi \to A\in D$$
4		fvmptd.4	$$⊢\phi \to C\in V$$
5		nfv	$$⊢Ⅎx\phi$$
6		nfcv	$$⊢\underset{\_}{Ⅎ}xA$$
7		nfcv	$$⊢\underset{\_}{Ⅎ}xC$$
8	1 2 3 4 5 6 7	fvmptdf	$$⊢\phi \to F\left(A\right)=C$$