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Theorem axext3 2447
 Description: A generalization of the Axiom of Extensionality in which and need not be distinct. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) Remove dependencies on ax-10 1786, ax-12 1803, ax-13 1968. (Revised by Wolf Lammen, 9-Dec-2019.)
Assertion
Ref Expression
axext3
Distinct variable groups:   ,   ,

Proof of Theorem axext3
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ax6ev 1721 . 2
2 elequ2 1772 . . . . . . 7
32bibi1d 319 . . . . . 6
43albidv 1689 . . . . 5
5 ax-ext 2445 . . . . 5
64, 5syl6bir 229 . . . 4
7 ax-7 1739 . . . 4
86, 7syld 44 . . 3
98exlimiv 1698 . 2
101, 9ax-mp 5 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184  wal 1377  wex 1596 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-ext 2445 This theorem depends on definitions:  df-bi 185  df-ex 1597 This theorem is referenced by:  axext4  2449  axextnd  8978  axextdist  29159
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