Metamath Proof Explorer
Description: Restricted existential specialization, using implicit substitution.
(Contributed by Glauco Siliprandi, 24-Dec-2020)
|
|
Ref |
Expression |
|
Hypotheses |
rspcef.1 |
⊢ Ⅎ 𝑥 𝜓 |
|
|
rspcef.2 |
⊢ Ⅎ 𝑥 𝐴 |
|
|
rspcef.3 |
⊢ Ⅎ 𝑥 𝐵 |
|
|
rspcef.4 |
⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) |
|
Assertion |
rspcef |
⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝜓 ) → ∃ 𝑥 ∈ 𝐵 𝜑 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
rspcef.1 |
⊢ Ⅎ 𝑥 𝜓 |
2 |
|
rspcef.2 |
⊢ Ⅎ 𝑥 𝐴 |
3 |
|
rspcef.3 |
⊢ Ⅎ 𝑥 𝐵 |
4 |
|
rspcef.4 |
⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) |
5 |
1 2 3 4
|
rspcegf |
⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝜓 ) → ∃ 𝑥 ∈ 𝐵 𝜑 ) |