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Mirrors > Home > MPE Home > Th. List > reximdva0 | Structured version Visualization version GIF version |
Description: Restricted existence deduced from nonempty class. (Contributed by NM, 1-Feb-2012.) |
Ref | Expression |
---|---|
reximdva0.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓) |
Ref | Expression |
---|---|
reximdva0 | ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 3931 | . . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
2 | reximdva0.1 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓) | |
3 | 2 | ex 450 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓)) |
4 | 3 | ancld 576 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∧ 𝜓))) |
5 | 4 | eximdv 1846 | . . . 4 ⊢ (𝜑 → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))) |
6 | 5 | imp 445 | . . 3 ⊢ ((𝜑 ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) |
7 | 1, 6 | sylan2b 492 | . 2 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) |
8 | df-rex 2918 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
9 | 7, 8 | sylibr 224 | 1 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∃wex 1704 ∈ wcel 1990 ≠ wne 2794 ∃wrex 2913 ∅c0 3915 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-rex 2918 df-v 3202 df-dif 3577 df-nul 3916 |
This theorem is referenced by: n0snor2el 4364 hashgt12el 13210 refun0 21318 cstucnd 22088 supxrnemnf 29534 kerunit 29823 elpaddn0 35086 |
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