Description: Equality theorem for restricted class abstractions. Inference version. (Contributed by GG, 1-Sep-2025)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | rabeqbii.1 | ⊢ 𝐴 = 𝐵 | |
| rabeqbii.2 | ⊢ ( 𝜑 ↔ 𝜓 ) | ||
| Assertion | rabeqbii | ⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑥 ∈ 𝐵 ∣ 𝜓 } |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rabeqbii.1 | ⊢ 𝐴 = 𝐵 | |
| 2 | rabeqbii.2 | ⊢ ( 𝜑 ↔ 𝜓 ) | |
| 3 | 1 | eleq2i | ⊢ ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) |
| 4 | 3 2 | anbi12i | ⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) ) |
| 5 | 4 | abbii | ⊢ { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } = { 𝑥 ∣ ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) } |
| 6 | df-rab | ⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } | |
| 7 | df-rab | ⊢ { 𝑥 ∈ 𝐵 ∣ 𝜓 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) } | |
| 8 | 5 6 7 | 3eqtr4i | ⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑥 ∈ 𝐵 ∣ 𝜓 } |