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Theorem nfsb4t 1887
Description: A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 1885). (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof rewritten by Jim Kingdon, 9-May-2018.)
Assertion
Ref Expression
nfsb4t (xzφ → (¬ z z = y → Ⅎz[y / x]φ))

Proof of Theorem nfsb4t
StepHypRef Expression
1 nfnf1 1433 . . . . 5 zzφ
21nfal 1465 . . . 4 zxzφ
3 nfnae 1607 . . . 4 z ¬ z z = y
42, 3nfan 1454 . . 3 z(xzφ ¬ z z = y)
5 df-nf 1347 . . . . . 6 (Ⅎzφz(φzφ))
65albii 1356 . . . . 5 (xzφxz(φzφ))
7 hbsb4t 1886 . . . . 5 (xz(φzφ) → (¬ z z = y → ([y / x]φz[y / x]φ)))
86, 7sylbi 114 . . . 4 (xzφ → (¬ z z = y → ([y / x]φz[y / x]φ)))
98imp 115 . . 3 ((xzφ ¬ z z = y) → ([y / x]φz[y / x]φ))
104, 9nfd 1413 . 2 ((xzφ ¬ z z = y) → Ⅎz[y / x]φ)
1110ex 108 1 (xzφ → (¬ z z = y → Ⅎz[y / x]φ))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wal 1240  wnf 1346  [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-in1 544  ax-in2 545  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-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643
This theorem is referenced by:  dvelimdf  1889
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