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

Metamath Proof Explorer

Theorem ralrid

Description: Sufficient condition for the restricted universal quantifier. Deduction form. (Contributed by Jonathan Ben-Naim, 3-Jun-2011)

Ref Expression
Hypothesis ralrid.1 ⊢ ( 𝜑 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) )
Assertion ralrid ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 )

Proof

Step Hyp Ref Expression
1 ralrid.1 ⊢ ( 𝜑 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) )
2 df-ral ⊢ ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) )
3 1 2 sylibr ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 )