Metamath Proof Explorer


Theorem ralss

Description: Restricted universal quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015)

Ref Expression
Assertion ralss ( 𝐴𝐵 → ( ∀ 𝑥𝐴 𝜑 ↔ ∀ 𝑥𝐵 ( 𝑥𝐴𝜑 ) ) )

Proof

Step Hyp Ref Expression
1 ssel ( 𝐴𝐵 → ( 𝑥𝐴𝑥𝐵 ) )
2 1 pm4.71rd ( 𝐴𝐵 → ( 𝑥𝐴 ↔ ( 𝑥𝐵𝑥𝐴 ) ) )
3 2 imbi1d ( 𝐴𝐵 → ( ( 𝑥𝐴𝜑 ) ↔ ( ( 𝑥𝐵𝑥𝐴 ) → 𝜑 ) ) )
4 impexp ( ( ( 𝑥𝐵𝑥𝐴 ) → 𝜑 ) ↔ ( 𝑥𝐵 → ( 𝑥𝐴𝜑 ) ) )
5 3 4 bitrdi ( 𝐴𝐵 → ( ( 𝑥𝐴𝜑 ) ↔ ( 𝑥𝐵 → ( 𝑥𝐴𝜑 ) ) ) )
6 5 ralbidv2 ( 𝐴𝐵 → ( ∀ 𝑥𝐴 𝜑 ↔ ∀ 𝑥𝐵 ( 𝑥𝐴𝜑 ) ) )