Description: When ps is always true in a context, a restricted class abstraction is equal to the restricting class. Deduction form of rabeqc . (Contributed by Steven Nguyen, 7-Jun-2023)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | rabeqcda.1 | ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝜓 ) | |
| Assertion | rabeqcda | ⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } = 𝐴 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rabeqcda.1 | ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝜓 ) | |
| 2 | df-rab | ⊢ { 𝑥 ∈ 𝐴 ∣ 𝜓 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) } | |
| 3 | 1 | ex | ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → 𝜓 ) ) |
| 4 | 3 | pm4.71d | ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) ) |
| 5 | 4 | bicomd | ⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ 𝑥 ∈ 𝐴 ) ) |
| 6 | 5 | abbi1dv | ⊢ ( 𝜑 → { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) } = 𝐴 ) |
| 7 | 2 6 | syl5eq | ⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } = 𝐴 ) |