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Theorem axextnd 8987
Description: A version of the Axiom of Extensionality with no distinct variable conditions. (Contributed by NM, 14-Aug-2003.)
Assertion
Ref Expression
axextnd

Proof of Theorem axextnd
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfnae 2058 . . . . . . . 8
2 nfnae 2058 . . . . . . . 8
31, 2nfan 1928 . . . . . . 7
4 nfcvf 2644 . . . . . . . . . 10
54adantr 465 . . . . . . . . 9
65nfcrd 2625 . . . . . . . 8
7 nfcvf 2644 . . . . . . . . . 10
87adantl 466 . . . . . . . . 9
98nfcrd 2625 . . . . . . . 8
106, 9nfbid 1933 . . . . . . 7
11 elequ1 1821 . . . . . . . . 9
12 elequ1 1821 . . . . . . . . 9
1311, 12bibi12d 321 . . . . . . . 8
1413a1i 11 . . . . . . 7
153, 10, 14cbvald 2025 . . . . . 6
16 axext3 2437 . . . . . 6
1715, 16syl6bir 229 . . . . 5
18 19.8a 1857 . . . . 5
1917, 18syl6 33 . . . 4
2019ex 434 . . 3
21 ax6e 2002 . . . . 5
22 ax-7 1790 . . . . . 6
2322aleximi 1653 . . . . 5
2421, 23mpi 17 . . . 4
2524a1d 25 . . 3
26 ax6e 2002 . . . . 5
27 ax-7 1790 . . . . . . 7
28 equcomi 1793 . . . . . . 7
2927, 28syl6 33 . . . . . 6
3029aleximi 1653 . . . . 5
3126, 30mpi 17 . . . 4
3231a1d 25 . . 3
3320, 25, 32pm2.61ii 165 . 2
343319.35ri 1690 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  E.wex 1612  F/_wnfc 2605
This theorem is referenced by:  zfcndext  9012  axextprim  29073  axextdfeq  29230  axextndbi  29237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-cleq 2449  df-clel 2452  df-nfc 2607
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