Description: Distribute restricted universal quantification over "or". (Contributed by Jeff Madsen, 19-Jun-2010) Shorten, reduce dv conditions. (Revised by Wolf Lammen, 20-Nov-2024)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | 2ralor | ⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜑 ∨ 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 ∨ ∀ 𝑦 ∈ 𝐵 𝜓 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | r19.32v | ⊢ ( ∀ 𝑦 ∈ 𝐵 ( 𝜑 ∨ 𝜓 ) ↔ ( 𝜑 ∨ ∀ 𝑦 ∈ 𝐵 𝜓 ) ) | |
| 2 | orcom | ⊢ ( ( 𝜑 ∨ ∀ 𝑦 ∈ 𝐵 𝜓 ) ↔ ( ∀ 𝑦 ∈ 𝐵 𝜓 ∨ 𝜑 ) ) | |
| 3 | 1 2 | bitri | ⊢ ( ∀ 𝑦 ∈ 𝐵 ( 𝜑 ∨ 𝜓 ) ↔ ( ∀ 𝑦 ∈ 𝐵 𝜓 ∨ 𝜑 ) ) |
| 4 | 3 | ralbii | ⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜑 ∨ 𝜓 ) ↔ ∀ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐵 𝜓 ∨ 𝜑 ) ) |
| 5 | r19.32v | ⊢ ( ∀ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐵 𝜓 ∨ 𝜑 ) ↔ ( ∀ 𝑦 ∈ 𝐵 𝜓 ∨ ∀ 𝑥 ∈ 𝐴 𝜑 ) ) | |
| 6 | orcom | ⊢ ( ( ∀ 𝑦 ∈ 𝐵 𝜓 ∨ ∀ 𝑥 ∈ 𝐴 𝜑 ) ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 ∨ ∀ 𝑦 ∈ 𝐵 𝜓 ) ) | |
| 7 | 4 5 6 | 3bitri | ⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜑 ∨ 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 ∨ ∀ 𝑦 ∈ 𝐵 𝜓 ) ) |