Description: Restricted quantifier version of 19.29 . See also r19.29r . (Contributed by NM, 31-Aug-1999) (Proof shortened by Andrew Salmon, 30-May-2011)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | r19.29 | ⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝜑 ∧ ∃ 𝑥 ∈ 𝐴 𝜓 ) → ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pm3.2 | ⊢ ( 𝜑 → ( 𝜓 → ( 𝜑 ∧ 𝜓 ) ) ) | |
| 2 | 1 | ralimi | ⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 → ∀ 𝑥 ∈ 𝐴 ( 𝜓 → ( 𝜑 ∧ 𝜓 ) ) ) |
| 3 | rexim | ⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜓 → ( 𝜑 ∧ 𝜓 ) ) → ( ∃ 𝑥 ∈ 𝐴 𝜓 → ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ) ) | |
| 4 | 2 3 | syl | ⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 → ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ) ) |
| 5 | 4 | imp | ⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝜑 ∧ ∃ 𝑥 ∈ 𝐴 𝜓 ) → ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ) |