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Metamath Proof Explorer
Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 10-Feb-1997) (Proof shortened by Wolf Lammen, 31-Oct-2024)
|
|
Ref |
Expression |
|
Hypothesis |
ralimia.1 |
⊢ ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝜓 ) ) |
|
Assertion |
reximia |
⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
ralimia.1 |
⊢ ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝜓 ) ) |
| 2 |
1
|
imdistani |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) |
| 3 |
2
|
reximi2 |
⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) |