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 41287 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 Ⅎwnfc 2963 ∃wrex 3141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rex 3146 df-v 3498 |
This theorem is referenced by: iinssdf 41415 opnvonmbllem1 42921 smfresal 43070 smfmullem2 43074 |
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