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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 4310 | . . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
2 | reximdva0.1 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓) | |
3 | 2 | ex 415 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓)) |
4 | 3 | ancld 553 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∧ 𝜓))) |
5 | 4 | eximdv 1914 | . . . 4 ⊢ (𝜑 → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))) |
6 | 5 | imp 409 | . . 3 ⊢ ((𝜑 ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) |
7 | 1, 6 | sylan2b 595 | . 2 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) |
8 | df-rex 3144 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
9 | 7, 8 | sylibr 236 | 1 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∃wex 1776 ∈ wcel 2110 ≠ wne 3016 ∃wrex 3139 ∅c0 4291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-rex 3144 df-dif 3939 df-nul 4292 |
This theorem is referenced by: n0snor2el 4758 hashgt12el 13777 refun0 22117 cstucnd 22887 supxrnemnf 30487 kerunit 30891 ssmxidllem 30973 elpaddn0 36930 |
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