Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbequ5 Structured version   Visualization version   GIF version

Theorem sbequ5 2480
 Description: Substitution does not change an identical variable specifier. Usage of this theorem is discouraged because it depends on ax-13 2382. (Contributed by NM, 15-May-1993.) (New usage is discouraged.)
Assertion
Ref Expression
sbequ5 ([𝑤 / 𝑧]∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑦)

Proof of Theorem sbequ5
StepHypRef Expression
1 nfae 2447 . 2 𝑧𝑥 𝑥 = 𝑦
21sbf 2270 1 ([𝑤 / 𝑧]∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑦)
 Colors of variables: wff setvar class Syntax hints:   ↔ 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-11 2159  ax-12 2176  ax-13 2382 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator