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Theorem axext4 2021
 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 2019. (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 1598 . . 3 (x = y → (z xz y))
21alrimiv 1751 . 2 (x = yz(z xz y))
3 axext3 2020 . 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 1240 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 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-nf 1347 This theorem is referenced by: (None)
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