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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 2161 | . 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
2 | 1 | nf5ri 2197 | 1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1540 [wsb 2074 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-10 2145 ax-12 2179 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-ex 1787 df-nf 1791 df-sb 2075 |
This theorem is referenced by: hbab1 2724 sb5ALT 41683 2sb5ndVD 42068 sb5ALTVD 42071 2sb5ndALT 42090 |
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