Metamath Proof Explorer


Theorem fvmptd3

Description: Deduction version of fvmpt . (Contributed by Glauco Siliprandi, 23-Oct-2021)

Ref Expression
Hypotheses fvmptd3.1 𝐹 = ( 𝑥𝐷𝐵 )
fvmptd3.2 ( 𝑥 = 𝐴𝐵 = 𝐶 )
fvmptd3.3 ( 𝜑𝐴𝐷 )
fvmptd3.4 ( 𝜑𝐶𝑉 )
Assertion fvmptd3 ( 𝜑 → ( 𝐹𝐴 ) = 𝐶 )

Proof

Step Hyp Ref Expression
1 fvmptd3.1 𝐹 = ( 𝑥𝐷𝐵 )
2 fvmptd3.2 ( 𝑥 = 𝐴𝐵 = 𝐶 )
3 fvmptd3.3 ( 𝜑𝐴𝐷 )
4 fvmptd3.4 ( 𝜑𝐶𝑉 )
5 2 1 fvmptg ( ( 𝐴𝐷𝐶𝑉 ) → ( 𝐹𝐴 ) = 𝐶 )
6 3 4 5 syl2anc ( 𝜑 → ( 𝐹𝐴 ) = 𝐶 )