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Theorem sbh 1637
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 1627 . . . 4 ([y / x]φx(x = y φ))
2 sbh.1 . . . . 5 (φxφ)
3219.41h 1553 . . . 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 1636 . . 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 1224  wex 1358  [wsb 1623
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 1312  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-4 1377  ax-i9 1400  ax-ial 1405
This theorem depends on definitions:  df-bi 110  df-sb 1624
This theorem is referenced by:  sbf  1638  sb6x  1640  nfs1f  1641  hbs1f  1642  sbid2h  1707  sblimv  1752  sbrim  1808  sbrbif  1814  elsb3  1830  elsb4  1831
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