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Metamath Proof Explorer
Description: Inference adding restricted existential quantifier to both sides of an
equivalence. (Contributed by NM, 26-Oct-1999)
|
|
Ref |
Expression |
|
Hypothesis |
rexbiia.1 |
⊢ ( 𝑥 ∈ 𝐴 → ( 𝜑 ↔ 𝜓 ) ) |
|
Assertion |
rexbiia |
⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ∈ 𝐴 𝜓 ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
rexbiia.1 |
⊢ ( 𝑥 ∈ 𝐴 → ( 𝜑 ↔ 𝜓 ) ) |
| 2 |
1
|
pm5.32i |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) |
| 3 |
2
|
rexbii2 |
⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ∈ 𝐴 𝜓 ) |