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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
Distinct variable groups:   ,   ,

Proof of Theorem axext4
StepHypRef Expression
1 elequ2 1715 . . 3
21alrimiv 1631 . 2
3 axext3 2336 . 2
42, 3impbii 180 1
Colors of variables: wff setvar class
Syntax hints:   wb 176  wal 1540   wceq 1642   wcel 1710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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)
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