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Metamath Proof Explorer
Description: Equivalent wff's yield equal restricted class abstractions (deduction
form). (Contributed by NM, 28-Nov-2003) (Proof shortened by SN, 3-Dec-2023)
|
|
Ref |
Expression |
|
Hypothesis |
rabbidva.1 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 ↔ 𝜒 ) ) |
|
Assertion |
rabbidva |
⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } = { 𝑥 ∈ 𝐴 ∣ 𝜒 } ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
rabbidva.1 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 ↔ 𝜒 ) ) |
| 2 |
1
|
pm5.32da |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜒 ) ) ) |
| 3 |
2
|
rabbidva2 |
⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } = { 𝑥 ∈ 𝐴 ∣ 𝜒 } ) |