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Metamath Proof Explorer

Theorem raleqi

Description: Equality inference for restricted universal quantifier. (Contributed by Paul Chapman, 22-Jun-2011)

Ref Expression
Hypothesis raleq1i.1 ⊢ 𝐴 = 𝐵
Assertion raleqi ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ∈ 𝐵 𝜑 )

Proof

Step Hyp Ref Expression
1 raleq1i.1 ⊢ 𝐴 = 𝐵
2 raleq ⊢ ( 𝐴 = 𝐵 → ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ∈ 𝐵 𝜑 ) )
3 1 2 ax-mp ⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ∈ 𝐵 𝜑 )