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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 2154 | . 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
2 | 1 | nf5ri 2193 | 1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1535 [wsb 2062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-10 2139 ax-12 2175 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1777 df-nf 1781 df-sb 2063 |
This theorem is referenced by: hbab1OLD 2722 sb5ALT 44523 2sb5ndVD 44908 sb5ALTVD 44911 2sb5ndALT 44930 |
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