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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 3714 al0ssb 5307 rext 5447 relop 5848 dfpo2 6292 frxp 8108 frxp2 8126 findcard 9159 pssnn 9164 ssfi 9169 pssnnOLD 9261 fiint 9320 marypha1lem 9424 dfom3 9638 elom3 9639 ttrclss 9711 aceq3lem 10111 dfac3 10112 dfac5lem4 10117 dfac8 10126 dfac9 10127 dfacacn 10132 dfac13 10133 kmlem1 10141 kmlem10 10150 fin23lem34 10337 fin23lem35 10338 zorn2lem7 10493 zornn0g 10496 axgroth6 10819 nnunb 12464 symggen 19332 gsumval3lem2 19768 gsumzaddlem 19783 dfac14 23113 i1fd 25189 chlimi 30474 zarclssn 32841 ddemeas 33222 dfon2lem4 34746 dfon2lem5 34747 dfon2lem7 34749 ttac 41760 dfac11 41789 dfac21 41793 setrec2fun 47690 |
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