MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax13dgen2 Structured version   Visualization version   GIF version

Theorem ax13dgen2 2133
Description: Degenerate instance of ax-13 2381 where bundled variables 𝑥 and 𝑧 have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
Assertion
Ref Expression
ax13dgen2 𝑥 = 𝑦 → (𝑦 = 𝑥 → ∀𝑥 𝑦 = 𝑥))

Proof of Theorem ax13dgen2
StepHypRef Expression
1 equcomi 2015 . 2 (𝑦 = 𝑥𝑥 = 𝑦)
2 pm2.21 123 . 2 𝑥 = 𝑦 → (𝑥 = 𝑦 → ∀𝑥 𝑦 = 𝑥))
31, 2syl5 34 1 𝑥 = 𝑦 → (𝑦 = 𝑥 → ∀𝑥 𝑦 = 𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator