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Mirrors > Home > ILE Home > Th. List > rspec | GIF version |
Description: Specialization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.) |
Ref | Expression |
---|---|
rspec.1 | ⊢ ∀𝑥 ∈ 𝐴 𝜑 |
Ref | Expression |
---|---|
rspec | ⊢ (𝑥 ∈ 𝐴 → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rspec.1 | . 2 ⊢ ∀𝑥 ∈ 𝐴 𝜑 | |
2 | rsp 2424 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (𝑥 ∈ 𝐴 → 𝜑)) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ (𝑥 ∈ 𝐴 → 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1439 ∀wral 2360 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-4 1446 |
This theorem depends on definitions: df-bi 116 df-ral 2365 |
This theorem is referenced by: rspec2 2463 vtoclri 2695 isarep2 5114 ecopover 6404 ecopoverg 6407 indstr 9142 |
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