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Mirrors > Home > NFE Home > Th. List > hbsb | GIF version |
Description: If z is not free in φ, it is not free in [y / x]φ when y and z are distinct. (Contributed by NM, 12-Aug-1993.) |
Ref | Expression |
---|---|
hbsb.1 | ⊢ (φ → ∀zφ) |
Ref | Expression |
---|---|
hbsb | ⊢ ([y / x]φ → ∀z[y / x]φ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbsb.1 | . . . 4 ⊢ (φ → ∀zφ) | |
2 | 1 | nfi 1551 | . . 3 ⊢ Ⅎzφ |
3 | 2 | nfsb 2109 | . 2 ⊢ Ⅎz[y / x]φ |
4 | 3 | nfri 1762 | 1 ⊢ ([y / x]φ → ∀z[y / x]φ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1540 [wsb 1648 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 |
This theorem depends on definitions: df-bi 177 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 |
This theorem is referenced by: hbab 2344 hblem 2457 |
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