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Metamath Proof Explorer
Description: Inference adding two restricted existential quantifiers to both sides of
an equivalence. (Contributed by NM, 11-Nov-1995)
|
|
Ref |
Expression |
|
Hypothesis |
2rexbii.1 |
⊢ ( 𝜑 ↔ 𝜓 ) |
|
Assertion |
2rexbii |
⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 ↔ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜓 ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
2rexbii.1 |
⊢ ( 𝜑 ↔ 𝜓 ) |
| 2 |
1
|
rexbii |
⊢ ( ∃ 𝑦 ∈ 𝐵 𝜑 ↔ ∃ 𝑦 ∈ 𝐵 𝜓 ) |
| 3 |
2
|
rexbii |
⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 ↔ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜓 ) |