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

Theorem axextb 2702
Description: A bidirectional version of the axiom of extensionality. Although this theorem looks like a definition of equality, it requires the axiom of extensionality for its proof under our axiomatization. See the comments for ax-ext 2699 and df-cleq 2720. (Contributed by NM, 14-Nov-2008.)
Assertion
Ref Expression
axextb (𝑥 = 𝑦 ↔ ∀𝑧(𝑧𝑥𝑧𝑦))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem axextb
StepHypRef Expression
1 elequ2g 2115 . 2 (𝑥 = 𝑦 → ∀𝑧(𝑧𝑥𝑧𝑦))
2 axextg 2701 . 2 (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
31, 2impbii 208 1 (𝑥 = 𝑦 ↔ ∀𝑧(𝑧𝑥𝑧𝑦))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wal 1532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775
This theorem is referenced by:  dfcleq  2721  axc11next  43843
  Copyright terms: Public domain W3C validator