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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 𝑥, 𝑦 and 𝑧 are distinct. Version of nfsb 2575 requiring more disjoint variables, but fewer axioms. (Contributed by Wolf Lammen, 7-Feb-2023.) |
Ref | Expression |
---|---|
nfsbv.nf | ⊢ Ⅎ𝑧𝜑 |
Ref | Expression |
---|---|
nfsbv | ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb6 2307 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
2 | nfv 2013 | . . . 4 ⊢ Ⅎ𝑧 𝑥 = 𝑦 | |
3 | nfsbv.nf | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
4 | 2, 3 | nfim 1999 | . . 3 ⊢ Ⅎ𝑧(𝑥 = 𝑦 → 𝜑) |
5 | 4 | nfal 2355 | . 2 ⊢ Ⅎ𝑧∀𝑥(𝑥 = 𝑦 → 𝜑) |
6 | 1, 5 | nfxfr 1952 | 1 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1654 Ⅎwnf 1882 [wsb 2067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-10 2192 ax-11 2207 ax-12 2220 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-ex 1879 df-nf 1883 df-sb 2068 |
This theorem is referenced by: 2sb8ev 2378 sb8euv 2687 |
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