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| 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 2156 | . 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
| 2 | 1 | nf5ri 2195 | 1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wal 1538 [wsb 2064 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-10 2141 ax-12 2177 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-nf 1784 df-sb 2065 | 
| This theorem is referenced by: hbab1OLD 2724 sb5ALT 44545 2sb5ndVD 44930 sb5ALTVD 44933 2sb5ndALT 44952 | 
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