Theorem fvmptd2

Description: Deduction version of fvmpt (where the definition of the mapping does not depend on the common antecedent ph ). (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Hypotheses	fvmptd2.1	$$⊢F=\left(x\in D⟼B\right)$$
	fvmptd2.2	$$⊢\left(\phi \wedge x=A\right)\to B=C$$
	fvmptd2.3	$$⊢\phi \to A\in D$$
	fvmptd2.4	$$⊢\phi \to C\in V$$
Assertion	fvmptd2	$$⊢\phi \to F\left(A\right)=C$$

Proof

Step	Hyp	Ref	Expression
1		fvmptd2.1	$$⊢F=\left(x\in D⟼B\right)$$
2		fvmptd2.2	$$⊢\left(\phi \wedge x=A\right)\to B=C$$
3		fvmptd2.3	$$⊢\phi \to A\in D$$
4		fvmptd2.4	$$⊢\phi \to C\in V$$
5	1	a1i	$$⊢\phi \to F=\left(x\in D⟼B\right)$$
6	5 2 3 4	fvmptd	$$⊢\phi \to F\left(A\right)=C$$