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Theorem hbsb4t 1886
Description: A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 1885). (Contributed by NM, 7-Apr-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Assertion
Ref Expression
hbsb4t (xz(φzφ) → (¬ z z = y → ([y / x]φz[y / x]φ)))

Proof of Theorem hbsb4t
StepHypRef Expression
1 hba1 1430 . . 3 (zφzzφ)
21hbsb4 1885 . 2 z z = y → ([y / x]zφz[y / x]zφ))
3 spsbim 1721 . . . . 5 (x(φzφ) → ([y / x]φ → [y / x]zφ))
43sps 1427 . . . 4 (zx(φzφ) → ([y / x]φ → [y / x]zφ))
5 ax-4 1397 . . . . . . 7 (zφφ)
65sbimi 1644 . . . . . 6 ([y / x]zφ → [y / x]φ)
76alimi 1341 . . . . 5 (z[y / x]zφz[y / x]φ)
87a1i 9 . . . 4 (zx(φzφ) → (z[y / x]zφz[y / x]φ))
94, 8imim12d 68 . . 3 (zx(φzφ) → (([y / x]zφz[y / x]zφ) → ([y / x]φz[y / x]φ)))
109a7s 1340 . 2 (xz(φzφ) → (([y / x]zφz[y / x]zφ) → ([y / x]φz[y / x]φ)))
112, 10syl5 28 1 (xz(φzφ) → (¬ z z = y → ([y / x]φz[y / x]φ)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  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-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-nf 1347  df-sb 1643
This theorem is referenced by:  nfsb4t  1887
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