Theorem fvmptd4

Description: Deduction version of fvmpt (where the substitution hypothesis does not have the antecedent ph ). (Contributed by SN, 26-Jul-2024)

	Ref	Expression
Hypotheses	fvmptd4.1	⊢ ( 𝑥 = 𝐴 → 𝐵 = 𝐶 )
	fvmptd4.2	⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐷 ↦ 𝐵 ) )
	fvmptd4.3	⊢ ( 𝜑 → 𝐴 ∈ 𝐷 )
	fvmptd4.4	⊢ ( 𝜑 → 𝐶 ∈ 𝑉 )
Assertion	fvmptd4	⊢ ( 𝜑 → ( 𝐹 ‘ 𝐴 ) = 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		fvmptd4.1	⊢ ( 𝑥 = 𝐴 → 𝐵 = 𝐶 )
2		fvmptd4.2	⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐷 ↦ 𝐵 ) )
3		fvmptd4.3	⊢ ( 𝜑 → 𝐴 ∈ 𝐷 )
4		fvmptd4.4	⊢ ( 𝜑 → 𝐶 ∈ 𝑉 )
5	1	adantl	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → 𝐵 = 𝐶 )
6	2 5 3 4	fvmptd	⊢ ( 𝜑 → ( 𝐹 ‘ 𝐴 ) = 𝐶 )