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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 3587 | . 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 1540 = wceq 1542 ∈ wcel 2107 Vcvv 3475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 |
This theorem is referenced by: morex 3716 al0ssb 5309 rext 5449 relop 5851 dfpo2 6296 frxp 8112 frxp2 8130 findcard 9163 pssnn 9168 ssfi 9173 pssnnOLD 9265 fiint 9324 marypha1lem 9428 dfom3 9642 elom3 9643 ttrclss 9715 aceq3lem 10115 dfac3 10116 dfac5lem4 10121 dfac8 10130 dfac9 10131 dfacacn 10136 dfac13 10137 kmlem1 10145 kmlem10 10154 fin23lem34 10341 fin23lem35 10342 zorn2lem7 10497 zornn0g 10500 axgroth6 10823 nnunb 12468 symggen 19338 gsumval3lem2 19774 gsumzaddlem 19789 dfac14 23122 i1fd 25198 chlimi 30487 zarclssn 32853 ddemeas 33234 dfon2lem4 34758 dfon2lem5 34759 dfon2lem7 34761 ttac 41775 dfac11 41804 dfac21 41808 setrec2fun 47737 |
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