Description: Deduction of restricted abstraction subclass from implication. (Contributed by Glauco Siliprandi, 21-Dec-2024)
Ref | Expression | ||
---|---|---|---|
Hypotheses | ss2rabdf.1 | ⊢ Ⅎ 𝑥 𝜑 | |
ss2rabdf.2 | ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 → 𝜒 ) ) | ||
Assertion | ss2rabdf | ⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } ⊆ { 𝑥 ∈ 𝐴 ∣ 𝜒 } ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ss2rabdf.1 | ⊢ Ⅎ 𝑥 𝜑 | |
2 | ss2rabdf.2 | ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 → 𝜒 ) ) | |
3 | 2 | ex | ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → ( 𝜓 → 𝜒 ) ) ) |
4 | 1 3 | ralrimi | ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 ( 𝜓 → 𝜒 ) ) |
5 | ss2rab | ⊢ ( { 𝑥 ∈ 𝐴 ∣ 𝜓 } ⊆ { 𝑥 ∈ 𝐴 ∣ 𝜒 } ↔ ∀ 𝑥 ∈ 𝐴 ( 𝜓 → 𝜒 ) ) | |
6 | 4 5 | sylibr | ⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } ⊆ { 𝑥 ∈ 𝐴 ∣ 𝜒 } ) |