MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  equsb1 Structured version   Visualization version   GIF version

Theorem equsb1 2490
Description: Substitution applied to an atomic wff. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker equsb1v 2103 if possible. (Contributed by NM, 10-May-1993.) (New usage is discouraged.)
Assertion
Ref Expression
equsb1 [𝑦 / 𝑥]𝑥 = 𝑦

Proof of Theorem equsb1
StepHypRef Expression
1 sb2 2478 . 2 (∀𝑥(𝑥 = 𝑦𝑥 = 𝑦) → [𝑦 / 𝑥]𝑥 = 𝑦)
2 id 22 . 2 (𝑥 = 𝑦𝑥 = 𝑦)
31, 2mpg 1799 1 [𝑦 / 𝑥]𝑥 = 𝑦
Colors of variables: wff setvar class
Syntax hints:  wi 4  [wsb 2067
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 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-12 2171  ax-13 2371
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ex 1782  df-nf 1786  df-sb 2068
This theorem is referenced by:  sbequ8  2500  sbie  2501  frege54cor1b  42630  sb5ALT  43271  sb5ALTVD  43659
  Copyright terms: Public domain W3C validator