Description: Theorem 19.29 of Margaris p. 90 with two restricted quantifiers, deduction version. (Contributed by Thierry Arnoux, 30-Jan-2017) (Proof shortened by Wolf Lammen, 4-Nov-2024)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | r19.29d2r.1 | ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜓 ) | |
| r19.29d2r.2 | ⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜒 ) | ||
| Assertion | r19.29d2r | ⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝜓 ∧ 𝜒 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | r19.29d2r.1 | ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜓 ) | |
| 2 | r19.29d2r.2 | ⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜒 ) | |
| 3 | 1 2 | jca | ⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜓 ∧ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜒 ) ) |
| 4 | 2r19.29 | ⊢ ( ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜓 ∧ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜒 ) → ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝜓 ∧ 𝜒 ) ) | |
| 5 | 3 4 | syl | ⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝜓 ∧ 𝜒 ) ) |