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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rspcef | Structured version Visualization version GIF version |
Description: Restricted existential specialization, using implicit substitution. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
Ref | Expression |
---|---|
rspcef.1 | ⊢ Ⅎ𝑥𝜓 |
rspcef.2 | ⊢ Ⅎ𝑥𝐴 |
rspcef.3 | ⊢ Ⅎ𝑥𝐵 |
rspcef.4 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
rspcef | ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rspcef.1 | . 2 ⊢ Ⅎ𝑥𝜓 | |
2 | rspcef.2 | . 2 ⊢ Ⅎ𝑥𝐴 | |
3 | rspcef.3 | . 2 ⊢ Ⅎ𝑥𝐵 | |
4 | rspcef.4 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
5 | 1, 2, 3, 4 | rspcegf 44960 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 Ⅎwnf 1779 ∈ wcel 2105 Ⅎwnfc 2887 ∃wrex 3067 |
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-cleq 2726 df-clel 2813 df-nfc 2889 df-rex 3068 |
This theorem is referenced by: iinssdf 45078 rspced 45109 opnvonmbllem1 46587 smfresal 46743 smfmullem2 46747 |
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