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

Theorem sbf 2026
 Description: Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
Hypothesis
Ref Expression
sbf.1 xφ
Assertion
Ref Expression
sbf ([y / x]φφ)

Proof of Theorem sbf
StepHypRef Expression
1 sbf.1 . 2 xφ
2 sbft 2025 . 2 (Ⅎxφ → ([y / x]φφ))
31, 2ax-mp 8 1 ([y / x]φφ)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176  Ⅎwnf 1544  [wsb 1648 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649 This theorem is referenced by:  sbh  2027  sbf2  2028  sb6x  2029  nfs1f  2030  sbequ5  2031  sbequ6  2032  sbt  2033  sbrim  2067  sblim  2068  sbrbif  2074  sbid2  2084  sbabel  2515
 Copyright terms: Public domain W3C validator