Metamath Proof Explorer
Description: A lemma for eliminating an existential quantifier. (Contributed by Giovanni Mascellani, 30-May-2019)
|
|
Ref |
Expression |
|
Hypotheses |
exlimddvf.1 |
⊢ ( 𝜑 → ∃ 𝑥 𝜃 ) |
|
|
exlimddvf.2 |
⊢ Ⅎ 𝑥 𝜓 |
|
|
exlimddvf.3 |
⊢ ( ( 𝜃 ∧ 𝜓 ) → 𝜒 ) |
|
|
exlimddvf.4 |
⊢ Ⅎ 𝑥 𝜒 |
|
Assertion |
exlimddvf |
⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
exlimddvf.1 |
⊢ ( 𝜑 → ∃ 𝑥 𝜃 ) |
2 |
|
exlimddvf.2 |
⊢ Ⅎ 𝑥 𝜓 |
3 |
|
exlimddvf.3 |
⊢ ( ( 𝜃 ∧ 𝜓 ) → 𝜒 ) |
4 |
|
exlimddvf.4 |
⊢ Ⅎ 𝑥 𝜒 |
5 |
3
|
expcom |
⊢ ( 𝜓 → ( 𝜃 → 𝜒 ) ) |
6 |
2 4 5
|
exlimd |
⊢ ( 𝜓 → ( ∃ 𝑥 𝜃 → 𝜒 ) ) |
7 |
1 6
|
mpan9 |
⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 ) |