Description: Inference quantifying antecedent, nested antecedent, and consequent, with a strong hypothesis. (Contributed by NM, 19-Dec-2006) Allow shortening of ralim . (Revised by Wolf Lammen, 1-Dec-2019)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | ral2imi.1 | ⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) ) | |
| Assertion | ral2imi | ⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 → ∀ 𝑥 ∈ 𝐴 𝜒 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ral2imi.1 | ⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) ) | |
| 2 | df-ral | ⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ) | |
| 3 | 1 | imim3i | ⊢ ( ( 𝑥 ∈ 𝐴 → 𝜑 ) → ( ( 𝑥 ∈ 𝐴 → 𝜓 ) → ( 𝑥 ∈ 𝐴 → 𝜒 ) ) ) |
| 4 | 3 | al2imi | ⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) → ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜒 ) ) ) |
| 5 | df-ral | ⊢ ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) ) | |
| 6 | df-ral | ⊢ ( ∀ 𝑥 ∈ 𝐴 𝜒 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜒 ) ) | |
| 7 | 4 5 6 | 3imtr4g | ⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) → ( ∀ 𝑥 ∈ 𝐴 𝜓 → ∀ 𝑥 ∈ 𝐴 𝜒 ) ) |
| 8 | 2 7 | sylbi | ⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 → ∀ 𝑥 ∈ 𝐴 𝜒 ) ) |