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Mirrors > Home > MPE Home > Th. List > nfsbv | Structured version Visualization version GIF version |
Description: If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑧 is distinct from 𝑥 and 𝑦. Version of nfsb 2565 requiring more disjoint variables, but fewer axioms. (Contributed by Mario Carneiro, 11-Aug-2016.) (Revised by Wolf Lammen, 7-Feb-2023.) Remove disjoint variable condition on 𝑥, 𝑦. (Revised by Steven Nguyen, 13-Aug-2023.) |
Ref | Expression |
---|---|
nfsbv.nf | ⊢ Ⅎ𝑧𝜑 |
Ref | Expression |
---|---|
nfsbv | ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sb 2070 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑤(𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤 → 𝜑))) | |
2 | nfv 1915 | . . . 4 ⊢ Ⅎ𝑧 𝑤 = 𝑦 | |
3 | nfv 1915 | . . . . . 6 ⊢ Ⅎ𝑧 𝑥 = 𝑤 | |
4 | nfsbv.nf | . . . . . 6 ⊢ Ⅎ𝑧𝜑 | |
5 | 3, 4 | nfim 1897 | . . . . 5 ⊢ Ⅎ𝑧(𝑥 = 𝑤 → 𝜑) |
6 | 5 | nfal 2342 | . . . 4 ⊢ Ⅎ𝑧∀𝑥(𝑥 = 𝑤 → 𝜑) |
7 | 2, 6 | nfim 1897 | . . 3 ⊢ Ⅎ𝑧(𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤 → 𝜑)) |
8 | 7 | nfal 2342 | . 2 ⊢ Ⅎ𝑧∀𝑤(𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤 → 𝜑)) |
9 | 1, 8 | nfxfr 1853 | 1 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1535 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-10 2145 ax-11 2161 ax-12 2177 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-nf 1785 df-sb 2070 |
This theorem is referenced by: hbsbw 2351 sbco2v 2352 2sb8ev 2375 sb8euv 2685 2mo 2733 nfcrii 2972 cbvralfw 3439 cbvreuw 3445 cbvrabw 3491 cbvrabcsfw 3926 cbvopab1 5141 cbvmptf 5167 ralxpf 5719 cbviotaw 6323 cbvriotaw 7125 dfoprab4f 7756 mo5f 30255 ax11-pm2 34161 dfich2 43620 dfich2ai 43621 ichbi12i 43625 |
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