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Theorem axext4 2388
 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 2385 and df-cleq 2397. (Contributed by NM, 14-Nov-2008.)
Assertion
Ref Expression
axext4
Distinct variable groups:   ,   ,

Proof of Theorem axext4
StepHypRef Expression
1 elequ2 1726 . . 3
21alrimiv 1638 . 2
3 axext3 2387 . 2
42, 3impbii 181 1
 Colors of variables: wff set class Syntax hints:   wb 177  wal 1546 This theorem is referenced by:  ax10ext  27474 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-11 1757  ax-12 1946  ax-ext 2385 This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-nf 1551
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