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Theorem hbsb4 1885
 Description: A variable not free remains so after substitution with a distinct variable. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.)
Hypothesis
Ref Expression
hbsb4.1 (φzφ)
Assertion
Ref Expression
hbsb4 z z = y → ([y / x]φz[y / x]φ))

Proof of Theorem hbsb4
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 hbsb4.1 . . 3 (φzφ)
21hbsb 1820 . 2 ([w / x]φz[w / x]φ)
3 sbequ 1718 . 2 (w = y → ([w / x]φ ↔ [y / x]φ))
42, 3dvelimALT 1883 1 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:  hbsb4t  1886  dvelimf  1888
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