Description: Restricted quantifier version of spesbcd . (Contributed by Eric Schmidt, 29-Sep-2025)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | rspesbcd.1 | ⊢ ( 𝜑 → 𝐴 ∈ 𝐵 ) | |
| rspesbcd.2 | ⊢ ( 𝜑 → [ 𝐴 / 𝑥 ] 𝜓 ) | ||
| Assertion | rspesbcd | ⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐵 𝜓 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rspesbcd.1 | ⊢ ( 𝜑 → 𝐴 ∈ 𝐵 ) | |
| 2 | rspesbcd.2 | ⊢ ( 𝜑 → [ 𝐴 / 𝑥 ] 𝜓 ) | |
| 3 | sbcel1v | ⊢ ( [ 𝐴 / 𝑥 ] 𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵 ) | |
| 4 | 1 3 | sylibr | ⊢ ( 𝜑 → [ 𝐴 / 𝑥 ] 𝑥 ∈ 𝐵 ) |
| 5 | sbcan | ⊢ ( [ 𝐴 / 𝑥 ] ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) ↔ ( [ 𝐴 / 𝑥 ] 𝑥 ∈ 𝐵 ∧ [ 𝐴 / 𝑥 ] 𝜓 ) ) | |
| 6 | 4 2 5 | sylanbrc | ⊢ ( 𝜑 → [ 𝐴 / 𝑥 ] ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) ) |
| 7 | 6 | spesbcd | ⊢ ( 𝜑 → ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) ) |
| 8 | df-rex | ⊢ ( ∃ 𝑥 ∈ 𝐵 𝜓 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) ) | |
| 9 | 7 8 | sylibr | ⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐵 𝜓 ) |