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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 2153 | . 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
2 | 1 | nf5ri 2188 | 1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1537 [wsb 2067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-10 2137 ax-12 2171 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ex 1783 df-nf 1787 df-sb 2068 |
This theorem is referenced by: hbab1OLD 2725 sb5ALT 42145 2sb5ndVD 42530 sb5ALTVD 42533 2sb5ndALT 42552 |
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