MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax13dgen4 Structured version   Visualization version   GIF version

Theorem ax13dgen4 2145
Description: Degenerate instance of ax-13 2392 where bundled variables 𝑥, 𝑦, and 𝑧 have a common substitution. Therefore, also a degenerate instance of ax13dgen1 2142, ax13dgen2 2143, and ax13dgen3 2144. Also an instance of the intuitionistic tautology pm2.21 123. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.) Reduce axiom usage. (Revised by Wolf Lammen, 10-Oct-2021.)
Assertion
Ref Expression
ax13dgen4 𝑥 = 𝑥 → (𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))

Proof of Theorem ax13dgen4
StepHypRef Expression
1 pm2.21 123 1 𝑥 = 𝑥 → (𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator