Description: Quantification restricted to a subclass. (Contributed by NM, 11-Mar-2006) Avoid axioms. (Revised by GG, 19-May-2025)
Ref | Expression | ||
---|---|---|---|
Assertion | ssralv | ⊢ ( 𝐴 ⊆ 𝐵 → ( ∀ 𝑥 ∈ 𝐵 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜑 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ss | ⊢ ( 𝐴 ⊆ 𝐵 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) ) | |
2 | imim1 | ⊢ ( ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) → ( ( 𝑥 ∈ 𝐵 → 𝜑 ) → ( 𝑥 ∈ 𝐴 → 𝜑 ) ) ) | |
3 | 2 | al2imi | ⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) → ( ∀ 𝑥 ( 𝑥 ∈ 𝐵 → 𝜑 ) → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ) ) |
4 | df-ral | ⊢ ( ∀ 𝑥 ∈ 𝐵 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐵 → 𝜑 ) ) | |
5 | df-ral | ⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ) | |
6 | 3 4 5 | 3imtr4g | ⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) → ( ∀ 𝑥 ∈ 𝐵 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜑 ) ) |
7 | 1 6 | sylbi | ⊢ ( 𝐴 ⊆ 𝐵 → ( ∀ 𝑥 ∈ 𝐵 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜑 ) ) |