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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 2382 with a disjoint variable condition, which does not require ax-13 2367. See spimvw 1992 for a version with two disjoint variable conditions, requiring fewer axioms, and spimv 2385 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 1966 | . . 3 ⊢ ∃𝑥 𝑥 = 𝑦 | |
3 | spimfv.1 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) | |
4 | 2, 3 | eximii 1832 | . 2 ⊢ ∃𝑥(𝜑 → 𝜓) |
5 | 1, 4 | 19.36i 2220 | 1 ⊢ (∀𝑥𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1532 Ⅎwnf 1778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-12 2167 |
This theorem depends on definitions: df-bi 206 df-ex 1775 df-nf 1779 |
This theorem is referenced by: chvarfv 2229 cbv3v2 2230 cbv3v 2327 setrec2fun 48123 |
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