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Mirrors > Home > ILE Home > Th. List > reximdva0m | GIF version |
Description: Restricted existence deduced from inhabited class. (Contributed by Jim Kingdon, 31-Jul-2018.) |
Ref | Expression |
---|---|
reximdva0m.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓) |
Ref | Expression |
---|---|
reximdva0m | ⊢ ((𝜑 ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reximdva0m.1 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓) | |
2 | 1 | ex 113 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓)) |
3 | 2 | ancld 318 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∧ 𝜓))) |
4 | 3 | eximdv 1801 | . . 3 ⊢ (𝜑 → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))) |
5 | 4 | imp 122 | . 2 ⊢ ((𝜑 ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) |
6 | df-rex 2354 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
7 | 5, 6 | sylibr 132 | 1 ⊢ ((𝜑 ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∃wex 1421 ∈ wcel 1433 ∃wrex 2349 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-4 1440 ax-17 1459 ax-ial 1467 |
This theorem depends on definitions: df-bi 115 df-rex 2354 |
This theorem is referenced by: (None) |
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