NFE Home New Foundations Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  NFE Home  >  Th. List  >  sbequ2 GIF version

Theorem sbequ2 1650
Description: An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sbequ2 (x = y → ([y / x]φφ))

Proof of Theorem sbequ2
StepHypRef Expression
1 df-sb 1649 . 2 ([y / x]φ ↔ ((x = yφ) x(x = y φ)))
2 simpl 443 . . 3 (((x = yφ) x(x = y φ)) → (x = yφ))
32com12 27 . 2 (x = y → (((x = yφ) x(x = y φ)) → φ))
41, 3syl5bi 208 1 (x = y → ([y / x]φφ))
Colors of variables: wff setvar class
Syntax hints:  wi 4   wa 358  wex 1541  [wsb 1648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 177  df-an 360  df-sb 1649
This theorem is referenced by:  stdpc7  1917  sbequ12  1919  dfsb2  2055  sbequi  2059  sbn  2062  sbi1  2063  mo  2226  mopick  2266
  Copyright terms: Public domain W3C validator