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Theorem sbh 1656
Description: Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 17-Oct-2004.)
Hypothesis
Ref Expression
sbh.1 (φxφ)
Assertion
Ref Expression
sbh ([y / x]φφ)

Proof of Theorem sbh
StepHypRef Expression
1 sb1 1646 . . . 4 ([y / x]φx(x = y φ))
2 sbh.1 . . . . 5 (φxφ)
3219.41h 1572 . . . 4 (x(x = y φ) ↔ (x x = y φ))
41, 3sylib 127 . . 3 ([y / x]φ → (x x = y φ))
54simprd 107 . 2 ([y / x]φφ)
6 stdpc4 1655 . . 3 (xφ → [y / x]φ)
72, 6syl 14 . 2 (φ → [y / x]φ)
85, 7impbii 117 1 ([y / x]φφ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  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-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-i9 1420  ax-ial 1424
This theorem depends on definitions:  df-bi 110  df-sb 1643
This theorem is referenced by:  sbf  1657  sb6x  1659  nfs1f  1660  hbs1f  1661  sbid2h  1726  sblimv  1771  sbrim  1827  sbrbif  1833  elsb3  1849  elsb4  1850
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