Description: Restricted quantifier version of 19.29r ; variation of r19.29 . (Contributed by NM, 31-Aug-1999) (Proof shortened by Wolf Lammen, 29-Jun-2023)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | r19.29r | ⊢ ( ( ∃ 𝑥 ∈ 𝐴 𝜑 ∧ ∀ 𝑥 ∈ 𝐴 𝜓 ) → ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | r19.29 | ⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝜓 ∧ ∃ 𝑥 ∈ 𝐴 𝜑 ) → ∃ 𝑥 ∈ 𝐴 ( 𝜓 ∧ 𝜑 ) ) | |
| 2 | 1 | ancoms | ⊢ ( ( ∃ 𝑥 ∈ 𝐴 𝜑 ∧ ∀ 𝑥 ∈ 𝐴 𝜓 ) → ∃ 𝑥 ∈ 𝐴 ( 𝜓 ∧ 𝜑 ) ) |
| 3 | pm3.22 | ⊢ ( ( 𝜓 ∧ 𝜑 ) → ( 𝜑 ∧ 𝜓 ) ) | |
| 4 | 3 | reximi | ⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜓 ∧ 𝜑 ) → ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ) |
| 5 | 2 4 | syl | ⊢ ( ( ∃ 𝑥 ∈ 𝐴 𝜑 ∧ ∀ 𝑥 ∈ 𝐴 𝜓 ) → ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ) |