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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 108 | . . . . 5 ⊢ (φ → (x ∈ A → ψ)) |
3 | 2 | ancld 308 | . . . 4 ⊢ (φ → (x ∈ A → (x ∈ A ∧ ψ))) |
4 | 3 | eximdv 1757 | . . 3 ⊢ (φ → (∃x x ∈ A → ∃x(x ∈ A ∧ ψ))) |
5 | 4 | imp 115 | . 2 ⊢ ((φ ∧ ∃x x ∈ A) → ∃x(x ∈ A ∧ ψ)) |
6 | df-rex 2306 | . 2 ⊢ (∃x ∈ A ψ ↔ ∃x(x ∈ A ∧ ψ)) | |
7 | 5, 6 | sylibr 137 | 1 ⊢ ((φ ∧ ∃x x ∈ A) → ∃x ∈ A ψ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∃wex 1378 ∈ wcel 1390 ∃wrex 2301 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-4 1397 ax-17 1416 ax-ial 1424 |
This theorem depends on definitions: df-bi 110 df-rex 2306 |
This theorem is referenced by: (None) |
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