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Metamath Proof Explorer
Description: Variation of 19.29 . (Contributed by NM, 18-Aug-1993) (Proof
shortened by Wolf Lammen, 12-Nov-2020)
|
|
Ref |
Expression |
|
Assertion |
19.29r |
⊢ ( ( ∃ 𝑥 𝜑 ∧ ∀ 𝑥 𝜓 ) → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
pm3.21 |
⊢ ( 𝜓 → ( 𝜑 → ( 𝜑 ∧ 𝜓 ) ) ) |
| 2 |
1
|
aleximi |
⊢ ( ∀ 𝑥 𝜓 → ( ∃ 𝑥 𝜑 → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) ) |
| 3 |
2
|
impcom |
⊢ ( ( ∃ 𝑥 𝜑 ∧ ∀ 𝑥 𝜓 ) → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) |