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Metamath Proof Explorer
Description: Deduction quantifying both antecedent and consequent, based on Theorem
19.22 of Margaris p. 90. (Contributed by NM, 22-May-1999)
|
|
Ref |
Expression |
|
Hypothesis |
ralimdva.1 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 → 𝜒 ) ) |
|
Assertion |
reximdva |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 → ∃ 𝑥 ∈ 𝐴 𝜒 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
ralimdva.1 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 → 𝜒 ) ) |
| 2 |
1
|
ex |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → ( 𝜓 → 𝜒 ) ) ) |
| 3 |
2
|
reximdvai |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 → ∃ 𝑥 ∈ 𝐴 𝜒 ) ) |