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

Theorem equsb1 2512
 Description: Substitution applied to an atomic wff. Usage of this theorem is discouraged because it depends on ax-13 2382. Use the weaker equsb1v 2110 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 2496 . 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 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  ax-13 2382 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070 This theorem is referenced by:  sbequ8  2523  sbie  2524  frege54cor1b  40582  sb5ALT  41218  sb5ALTVD  41606
 Copyright terms: Public domain W3C validator