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Metamath Proof Explorer

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 )