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Theorem hbsbv 1814
Description: This is a version of hbsb 1820 with an extra distinct variable constraint, on z and x. (Contributed by Jim Kingdon, 25-Dec-2017.)
Hypothesis
Ref Expression
hbsbv.1 (φzφ)
Assertion
Ref Expression
hbsbv ([y / x]φz[y / x]φ)
Distinct variable groups:   x,z   y,z
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem hbsbv
StepHypRef Expression
1 df-sb 1643 . 2 ([y / x]φ ↔ ((x = yφ) x(x = y φ)))
2 ax-17 1416 . . . 4 (x = yz x = y)
3 hbsbv.1 . . . 4 (φzφ)
42, 3hbim 1434 . . 3 ((x = yφ) → z(x = yφ))
52, 3hban 1436 . . . 4 ((x = y φ) → z(x = y φ))
65hbex 1524 . . 3 (x(x = y φ) → zx(x = y φ))
74, 6hban 1436 . 2 (((x = yφ) x(x = y φ)) → z((x = yφ) x(x = y φ)))
81, 7hbxfrbi 1358 1 ([y / x]φz[y / x]φ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1240  wex 1378  [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-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-i5r 1425
This theorem depends on definitions:  df-bi 110  df-sb 1643
This theorem is referenced by:  sbco2vlem  1817  2sb5rf  1862  2sb6rf  1863
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