MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  equsb3lem Structured version   Visualization version   GIF version

Theorem equsb3lem 2430
Description: Lemma for equsb3 2431. (Contributed by Raph Levien and FL, 4-Dec-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
Assertion
Ref Expression
equsb3lem ([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧)
Distinct variable groups:   𝑦,𝑧   𝑥,𝑦

Proof of Theorem equsb3lem
StepHypRef Expression
1 nfv 1840 . 2 𝑦 𝑥 = 𝑧
2 equequ1 1949 . 2 (𝑦 = 𝑥 → (𝑦 = 𝑧𝑥 = 𝑧))
31, 2sbie 2407 1 ([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧)
Colors of variables: wff setvar class
Syntax hints:  wb 196  [wsb 1877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707  df-sb 1878
This theorem is referenced by:  equsb3  2431  equsb3ALT  2432
  Copyright terms: Public domain W3C validator