Metamath Proof Explorer
Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 10-Feb-1997)
|
|
Ref |
Expression |
|
Hypothesis |
reximia.1 |
⊢ ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝜓 ) ) |
|
Assertion |
reximia |
⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
reximia.1 |
⊢ ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝜓 ) ) |
2 |
|
rexim |
⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) → ( ∃ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) ) |
3 |
2 1
|
mprg |
⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) |