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.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
2 | rspcef.3 | . . . . . 6 ⊢ Ⅎ𝑥𝐵 | |
3 | 1, 2 | nfel 2763 | . . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵 |
4 | rspcef.1 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
5 | 3, 4 | nfan 1816 | . . . 4 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐵 ∧ 𝜓) |
6 | eleq1 2676 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
7 | rspcef.4 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
8 | 6, 7 | anbi12d 743 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ (𝐴 ∈ 𝐵 ∧ 𝜓))) |
9 | 1, 5, 8 | spcegf 3262 | . . 3 ⊢ (𝐴 ∈ 𝐵 → ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))) |
10 | 9 | anabsi5 854 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)) |
11 | df-rex 2902 | . 2 ⊢ (∃𝑥 ∈ 𝐵 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)) | |
12 | 10, 11 | sylibr 223 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∃wex 1695 Ⅎwnf 1699 ∈ wcel 1977 Ⅎwnfc 2738 ∃wrex 2897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-v 3175 |
This theorem is referenced by: opnvonmbllem1 39522 smfresal 39673 smfmullem2 39677 |
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