Metamath Proof Explorer

Theorem raleleq

Description: All elements of a class are elements of a class equal to this class. (Contributed by AV, 30-Oct-2020)

Ref Expression
Assertion raleleq ( 𝐴 = 𝐵 → ∀ 𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 )

Proof

Step Hyp Ref Expression
1 eleq2 ⊢ ( 𝐴 = 𝐵 → ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) )
2 1 biimpd ⊢ ( 𝐴 = 𝐵 → ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) )
3 2 ralrimiv ⊢ ( 𝐴 = 𝐵 → ∀ 𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 )