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| Mirrors > Home > MPE Home > Th. List > rspcv | Structured version Visualization version GIF version | ||
| Description: Restricted specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) Drop ax-10 2141, ax-11 2157, ax-12 2177. (Revised by SN, 12-Dec-2023.) |
| Ref | Expression |
|---|---|
| rspcv.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
| Ref | Expression |
|---|---|
| rspcv | ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | id 22 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ 𝐵) | |
| 2 | rspcv.1 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
| 3 | 2 | adantl 481 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴) → (𝜑 ↔ 𝜓)) |
| 4 | 1, 3 | rspcdv 3614 | 1 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓)) |
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