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Theorem hbsb 2110
 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.)
Hypothesis
Ref Expression
hbsb.1 (φzφ)
Assertion
Ref Expression
hbsb ([y / x]φz[y / x]φ)
Distinct variable group:   y,z
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem hbsb
StepHypRef Expression
1 hbsb.1 . . . 4 (φzφ)
21nfi 1551 . . 3 zφ
32nfsb 2109 . 2 z[y / x]φ
43nfri 1762 1 ([y / x]φz[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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