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Theorem axc11n 2105
 Description: Derive set.mm's original ax-c11n 32169 from others. Commutation law for identical variable specifiers. The antecedent and consequent are true when and are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). Use aecom 2107 instead when this does not lengthen the proof. (Contributed by NM, 10-May-1993.) (Revised by NM, 7-Nov-2015.) (Proof shortened by Wolf Lammen, 6-Mar-2018.) Adapt to a modification of axc11nlem 1996. (Revised by Wolf Lammen, 30-Nov-2019.)
Assertion
Ref Expression
axc11n

Proof of Theorem axc11n
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ax6ev 1799 . . 3
2 equcomi 1845 . . . . . 6
3 dveeq1 2100 . . . . . 6
42, 3syl5com 31 . . . . 5
5 axc112 1995 . . . . . 6
63axc11nlem 1996 . . . . . 6
75, 6syl6 34 . . . . 5
84, 7syl9 73 . . . 4
98exlimiv 1769 . . 3
101, 9ax-mp 5 . 2
1110pm2.18d 114 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4  wal 1435  wex 1659 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-12 1907  ax-13 2055 This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-nf 1664 This theorem is referenced by:  aecom  2107  axi10  2404  wl-ax11-lem3  31621  wl-ax11-lem8  31626  2sb5ndVD  36947  e2ebindVD  36949  e2ebindALT  36966  2sb5ndALT  36969
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