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| Mirrors > Home > MPE Home > Th. List > spimfv | Structured version Visualization version GIF version | ||
| Description: Specialization, using implicit substitution. Version of spim 2392 with a disjoint variable condition, which does not require ax-13 2377. See spimvw 1995 for a version with two disjoint variable conditions, requiring fewer axioms, and spimv 2395 for another variant. (Contributed by NM, 10-Jan-1993.) (Revised by BJ, 31-May-2019.) |
| Ref | Expression |
|---|---|
| spimfv.nf | ⊢ Ⅎ𝑥𝜓 |
| spimfv.1 | ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
| Ref | Expression |
|---|---|
| spimfv | ⊢ (∀𝑥𝜑 → 𝜓) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | spimfv.nf | . 2 ⊢ Ⅎ𝑥𝜓 | |
| 2 | ax6ev 1969 | . . 3 ⊢ ∃𝑥 𝑥 = 𝑦 | |
| 3 | spimfv.1 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) | |
| 4 | 2, 3 | eximii 1837 | . 2 ⊢ ∃𝑥(𝜑 → 𝜓) |
| 5 | 1, 4 | 19.36i 2231 | 1 ⊢ (∀𝑥𝜑 → 𝜓) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∀wal 1538 Ⅎwnf 1783 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-ex 1780 df-nf 1784 |
| This theorem is referenced by: chvarfv 2240 cbv3v2 2241 cbv3v 2337 setrec2fun 49211 |
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