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Theorem axext4 2006
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 2004. (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 1583 . . 3 (x = y → (z xz y))
21alrimiv 1736 . 2 (x = yz(z xz y))
3 axext3 2005 . 2 (z(z xz y) → x = y)
42, 3impbii 117 1 (x = yz(z xz y))
Colors of variables: wff set class
Syntax hints:  wb 98  wal 1226
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1316  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-nf 1330
This theorem is referenced by: (None)
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