![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > hbs1 | Structured version Visualization version GIF version |
Description: The setvar 𝑥 is not free in [𝑦 / 𝑥]𝜑 when 𝑥 and 𝑦 are distinct. (Contributed by NM, 26-May-1993.) |
Ref | Expression |
---|---|
hbs1 | ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfs1v 2274 | . 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
2 | 1 | nf5ri 2219 | 1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1629 [wsb 2049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-10 2174 ax-12 2203 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-ex 1853 df-nf 1858 df-sb 2050 |
This theorem is referenced by: nfs1vOLD 2578 hbab1 2760 sb5ALT 39256 2sb5ndVD 39668 sb5ALTVD 39671 2sb5ndALT 39690 |
Copyright terms: Public domain | W3C validator |