Metamath Proof Explorer
Description: Expand a restricted existential quantifier to primitives. (Contributed by Rohan Ridenour, 13-Aug-2023)
|
|
Ref |
Expression |
|
Hypothesis |
expandrex.1 |
⊢ ( 𝜑 ↔ 𝜓 ) |
|
Assertion |
expandrex |
⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ¬ 𝜓 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
expandrex.1 |
⊢ ( 𝜑 ↔ 𝜓 ) |
2 |
|
notnotb |
⊢ ( 𝜓 ↔ ¬ ¬ 𝜓 ) |
3 |
1 2
|
bitri |
⊢ ( 𝜑 ↔ ¬ ¬ 𝜓 ) |
4 |
3
|
expandrexn |
⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ¬ 𝜓 ) ) |