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Theorem axext4 2280
 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 2277 and df-cleq 2289. (Contributed by NM, 14-Nov-2008.)
Assertion
Ref Expression
axext4
Distinct variable groups:   ,   ,

Proof of Theorem axext4
StepHypRef Expression
1 elequ2 1701 . . 3
21alrimiv 1621 . 2
3 axext3 2279 . 2
42, 3impbii 180 1
 Colors of variables: wff set class Syntax hints:   wb 176  wal 1530   wceq 1632   wcel 1696 This theorem is referenced by:  ax10ext  27709 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535
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