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Mirrors > Home > MPE Home > Th. List > rspec | Structured version Visualization version 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 3207 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (𝑥 ∈ 𝐴 → 𝜑)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝑥 ∈ 𝐴 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ∀wral 3140 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-12 2177 |
This theorem depends on definitions: df-bi 209 df-ex 1781 df-ral 3145 |
This theorem is referenced by: rspec2 3213 vtoclri 3587 wfis 6186 wfis2f 6188 wfis2 6190 isarep2 6445 mpoexw 7778 ecopover 8403 alephsuc2 9508 indstr 12319 reltxrnmnf 12738 ackbijnn 15185 mrelatglb0 17797 0frgp 18907 iccpnfcnv 23550 frins 33090 frins2f 33092 prter2 36019 |
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