 New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  axext4 GIF version

Theorem axext4 2337
 Description: A bidirectional version of Extensionality. Although this theorem "looks" like it is just a definition of equality, it requires the Axiom of Extensionality for its proof under our axiomatization. See the comments for ax-ext 2334 and df-cleq 2346. (Contributed by NM, 14-Nov-2008.)
Assertion
Ref Expression
axext4 (x = yz(z xz y))
Distinct variable groups:   x,z   y,z

Proof of Theorem axext4
StepHypRef Expression
1 elequ2 1715 . . 3 (x = y → (z xz y))
21alrimiv 1631 . 2 (x = yz(z xz y))
3 axext3 2336 . 2 (z(z xz y) → x = y)
42, 3impbii 180 1 (x = yz(z xz y))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator