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Mirrors > Home > MPE Home > Th. List > spcv | Structured version Visualization version GIF version |
Description: Rule of specialization, using implicit substitution. (Contributed by NM, 22-Jun-1994.) |
Ref | Expression |
---|---|
spcv.1 | ⊢ 𝐴 ∈ V |
spcv.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
spcv | ⊢ (∀𝑥𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spcv.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | spcv.2 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 2 | spcgv 3586 | . 2 ⊢ (𝐴 ∈ V → (∀𝑥𝜑 → 𝜓)) |
4 | 1, 3 | ax-mp 5 | 1 ⊢ (∀𝑥𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1539 = wceq 1541 ∈ wcel 2106 Vcvv 3474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-v 3476 |
This theorem is referenced by: morex 3715 al0ssb 5308 rext 5448 relop 5850 dfpo2 6295 frxp 8114 frxp2 8132 findcard 9165 pssnn 9170 ssfi 9175 pssnnOLD 9267 fiint 9326 marypha1lem 9430 dfom3 9644 elom3 9645 ttrclss 9717 aceq3lem 10117 dfac3 10118 dfac5lem4 10123 dfac8 10132 dfac9 10133 dfacacn 10138 dfac13 10139 kmlem1 10147 kmlem10 10156 fin23lem34 10343 fin23lem35 10344 zorn2lem7 10499 zornn0g 10502 axgroth6 10825 nnunb 12470 symggen 19340 gsumval3lem2 19776 gsumzaddlem 19791 dfac14 23129 i1fd 25205 chlimi 30525 zarclssn 32922 ddemeas 33303 dfon2lem4 34833 dfon2lem5 34834 dfon2lem7 34836 ttac 41863 dfac11 41892 dfac21 41896 setrec2fun 47821 |
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