Description: Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010) Reduce axiom usage. (Revised by GG, 2-Nov-2024)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | ralab.1 | ⊢ ( 𝑦 = 𝑥 → ( 𝜑 ↔ 𝜓 ) ) | |
| Assertion | ralab | ⊢ ( ∀ 𝑥 ∈ { 𝑦 ∣ 𝜑 } 𝜒 ↔ ∀ 𝑥 ( 𝜓 → 𝜒 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ralab.1 | ⊢ ( 𝑦 = 𝑥 → ( 𝜑 ↔ 𝜓 ) ) | |
| 2 | df-ral | ⊢ ( ∀ 𝑥 ∈ { 𝑦 ∣ 𝜑 } 𝜒 ↔ ∀ 𝑥 ( 𝑥 ∈ { 𝑦 ∣ 𝜑 } → 𝜒 ) ) | |
| 3 | df-clab | ⊢ ( 𝑥 ∈ { 𝑦 ∣ 𝜑 } ↔ [ 𝑥 / 𝑦 ] 𝜑 ) | |
| 4 | 1 | sbievw | ⊢ ( [ 𝑥 / 𝑦 ] 𝜑 ↔ 𝜓 ) |
| 5 | 3 4 | bitri | ⊢ ( 𝑥 ∈ { 𝑦 ∣ 𝜑 } ↔ 𝜓 ) |
| 6 | 5 | imbi1i | ⊢ ( ( 𝑥 ∈ { 𝑦 ∣ 𝜑 } → 𝜒 ) ↔ ( 𝜓 → 𝜒 ) ) |
| 7 | 6 | albii | ⊢ ( ∀ 𝑥 ( 𝑥 ∈ { 𝑦 ∣ 𝜑 } → 𝜒 ) ↔ ∀ 𝑥 ( 𝜓 → 𝜒 ) ) |
| 8 | 2 7 | bitri | ⊢ ( ∀ 𝑥 ∈ { 𝑦 ∣ 𝜑 } 𝜒 ↔ ∀ 𝑥 ( 𝜓 → 𝜒 ) ) |