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Theorem axext3 2020
Description: A generalization of the Axiom of Extensionality in which x and y need not be distinct. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
Assertion
Ref Expression
axext3 (z(z xz y) → x = y)
Distinct variable groups:   x,z   y,z

Proof of Theorem axext3
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 elequ2 1598 . . . . 5 (w = x → (z wz x))
21bibi1d 222 . . . 4 (w = x → ((z wz y) ↔ (z xz y)))
32albidv 1702 . . 3 (w = x → (z(z wz y) ↔ z(z xz y)))
4 equequ1 1595 . . 3 (w = x → (w = yx = y))
53, 4imbi12d 223 . 2 (w = x → ((z(z wz y) → w = y) ↔ (z(z xz y) → x = y)))
6 ax-ext 2019 . 2 (z(z wz y) → w = y)
75, 6chvarv 1809 1 (z(z xz y) → x = y)
Colors of variables: wff set class
Syntax hints:  wi 4  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:  axext4  2021
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