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Theorem hbsb 1820
 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.) (Proof rewritten by Jim Kingdon, 22-Mar-2018.)
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 1348 . . 3 zφ
32nfsb 1819 . 2 z[y / x]φ
43nfri 1409 1 ([y / x]φz[y / x]φ)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1240  [wsb 1642 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643 This theorem is referenced by:  sb10f  1868  hbsb4  1885  sb8euh  1920  hbab  2028  hblem  2142
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