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Mirrors > Home > MPE Home > Th. List > Mathboxes > rspesbcd | Structured version Visualization version GIF version |
Description: Restricted quantifier version of spesbcd 3891. (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 3861 | . . . . 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 3843 | . . . 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 3891 | . 2 ⊢ (𝜑 → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜓)) |
8 | df-rex 3068 | . 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 1775 ∈ wcel 2105 ∃wrex 3067 [wsbc 3790 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1539 df-ex 1776 df-nf 1780 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ral 3059 df-rex 3068 df-v 3479 df-sbc 3791 |
This theorem is referenced by: modelaxreplem3 44944 |
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