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Mirrors > Home > MPE Home > Th. List > sbf | Structured version Visualization version GIF version |
Description: Substitution for a variable not free in a wff does not affect it. For a version requiring disjoint variables but fewer axioms, see sbv 2098. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) |
Ref | Expression |
---|---|
sbf.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
sbf | ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbf.1 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | sbft 2270 | . 2 ⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 Ⅎwnf 1784 [wsb 2069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-12 2177 |
This theorem depends on definitions: df-bi 209 df-ex 1781 df-nf 1785 df-sb 2070 |
This theorem is referenced by: sbf2 2272 sbh 2273 nfs1f 2275 sbbibOLD 2289 sbrim 2313 sblim 2315 sbrbif 2321 sbievOLD 2331 sb6x 2487 sbequ5 2488 sbequ6 2489 sb2ae 2536 sbie 2544 sbid2 2550 sbabel 3015 nfcdeq 3768 mo5f 30253 suppss2f 30384 fmptdF 30401 disjdsct 30438 esumpfinvalf 31335 bj-sbf3 34162 bj-sbf4 34163 ellimcabssub0 41918 2reu8i 43332 ichf 43630 |
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