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Mirrors > Home > MPE Home > Th. List > spv | Structured version Visualization version GIF version |
Description: Specialization, using implicit substitution. (Contributed by NM, 30-Aug-1993.) |
Ref | Expression |
---|---|
spv.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
spv | ⊢ (∀𝑥𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spv.1 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
2 | 1 | biimpd 218 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
3 | 2 | spimv 2245 | 1 ⊢ (∀𝑥𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 ax-13 2234 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 |
This theorem is referenced by: chvarv 2251 cbvalv 2261 ru 3401 nalset 4723 isowe2 6500 tfisi 6950 findcard2 8085 marypha1lem 8222 setind 8493 karden 8641 kmlem4 8858 axgroth3 9532 ramcl 15571 alexsubALTlem3 21663 i1fd 23254 dfpo2 30898 dfon2lem6 30937 trer 31480 axc11n-16 33241 elsetrecslem 42243 |
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