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 41157 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 Ⅎwnf 1775 ∈ wcel 2105 Ⅎwnfc 2958 ∃wrex 3136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-v 3494 |
This theorem is referenced by: iinssdf 41284 opnvonmbllem1 42791 smfresal 42940 smfmullem2 42944 |
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