This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem fvmptd3

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

Ref Expression
Hypotheses fvmptd3.1 ⊢ F = x ∈ D ⟼ B
fvmptd3.2 ⊢ x = A → B = C
fvmptd3.3 ⊢ φ → A ∈ D
fvmptd3.4 ⊢ φ → C ∈ V
Assertion fvmptd3 ⊢ φ → F ⁡ A = C

Proof

Step Hyp Ref Expression
1 fvmptd3.1 ⊢ F = x ∈ D ⟼ B
2 fvmptd3.2 ⊢ x = A → B = C
3 fvmptd3.3 ⊢ φ → A ∈ D
4 fvmptd3.4 ⊢ φ → C ∈ V
5 2 1 fvmptg ⊢ A ∈ D ∧ C ∈ V → F ⁡ A = C
6 3 4 5 syl2anc ⊢ φ → F ⁡ A = C