MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbh Structured version   Visualization version   GIF version

Theorem sbh 2272
Description: Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 14-May-1993.)
Hypothesis
Ref Expression
sbh.1 (𝜑 → ∀𝑥𝜑)
Assertion
Ref Expression
sbh ([𝑦 / 𝑥]𝜑𝜑)

Proof of Theorem sbh
StepHypRef Expression
1 sbh.1 . . 3 (𝜑 → ∀𝑥𝜑)
21nf5i 2148 . 2 𝑥𝜑
32sbf 2270 1 ([𝑦 / 𝑥]𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536  [wsb 2069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2143  ax-12 2176
This theorem depends on definitions:  df-bi 210  df-ex 1782  df-nf 1786  df-sb 2070
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator