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Mirrors > Home > MPE Home > Th. List > spv | Structured version Visualization version GIF version |
Description: Specialization, using implicit substitution. See spvv 1956 for a version using fewer axioms. (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 221 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
3 | 2 | spimv 2322 | 1 ⊢ (∀𝑥𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∀wal 1506 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-12 2107 ax-13 2302 |
This theorem depends on definitions: df-bi 199 df-an 388 df-ex 1744 df-nf 1748 |
This theorem is referenced by: cbvalvOLD 2334 isowe2 6924 tfisi 7387 findcard2 8551 marypha1lem 8690 setind 8968 karden 9116 axgroth3 10049 ramcl 16219 alexsubALTlem3 22376 i1fd 24000 dfpo2 32548 dfon2lem6 32590 trer 33222 axc11n-16 35556 elsetrecslem 44202 |
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