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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rspesbcd | Structured version Visualization version GIF version | ||
| Description: Restricted quantifier version of spesbcd 3882. (Contributed by Eric Schmidt, 29-Sep-2025.) |
| Ref | Expression |
|---|---|
| rspesbcd.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
| rspesbcd.2 | ⊢ (𝜑 → [𝐴 / 𝑥]𝜓) |
| Ref | Expression |
|---|---|
| rspesbcd | ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rspesbcd.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
| 2 | sbcel1v 3855 | . . . . 5 ⊢ ([𝐴 / 𝑥]𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵) | |
| 3 | 1, 2 | sylibr 234 | . . . 4 ⊢ (𝜑 → [𝐴 / 𝑥]𝑥 ∈ 𝐵) |
| 4 | rspesbcd.2 | . . . 4 ⊢ (𝜑 → [𝐴 / 𝑥]𝜓) | |
| 5 | sbcan 3837 | . . . 4 ⊢ ([𝐴 / 𝑥](𝑥 ∈ 𝐵 ∧ 𝜓) ↔ ([𝐴 / 𝑥]𝑥 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜓)) | |
| 6 | 3, 4, 5 | sylanbrc 583 | . . 3 ⊢ (𝜑 → [𝐴 / 𝑥](𝑥 ∈ 𝐵 ∧ 𝜓)) |
| 7 | 6 | spesbcd 3882 | . 2 ⊢ (𝜑 → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜓)) |
| 8 | df-rex 3070 | . 2 ⊢ (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜓)) | |
| 9 | 7, 8 | sylibr 234 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∃wex 1779 ∈ wcel 2108 ∃wrex 3069 [wsbc 3787 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-v 3481 df-sbc 3788 |
| This theorem is referenced by: modelaxreplem3 44988 |
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