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Metamath Proof Explorer
Description: Formula-building rule for restricted existential uniqueness quantifier
(deduction form). (Contributed by NM, 17-Oct-1996)
|
|
Ref |
Expression |
|
Hypothesis |
rmobidv.1 |
⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) |
|
Assertion |
reubidv |
⊢ ( 𝜑 → ( ∃! 𝑥 ∈ 𝐴 𝜓 ↔ ∃! 𝑥 ∈ 𝐴 𝜒 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
rmobidv.1 |
⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) |
| 2 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 ↔ 𝜒 ) ) |
| 3 |
2
|
reubidva |
⊢ ( 𝜑 → ( ∃! 𝑥 ∈ 𝐴 𝜓 ↔ ∃! 𝑥 ∈ 𝐴 𝜒 ) ) |