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Theorem ax9 1893
 Description: Theorem showing that ax-9 1638 follows from the weaker version ax9v 1639. (Even though this theorem depends on ax-9 1638, all references of ax-9 1638 are made via ax9v 1639. An earlier version stated ax9v 1639 as a separate axiom, but having two axioms caused some confusion.) This theorem should be referenced in place of ax-9 1638 so that all proofs can be traced back to ax9v 1639. (Contributed by NM, 12-Nov-2013.) (Revised by NM, 25-Jul-2015.)
Assertion
Ref Expression
ax9
Dummy variable is distinct from all other variables.

Proof of Theorem ax9
StepHypRef Expression
1 sp 1719 . . 3
2 sp 1719 . . 3
31, 2nsyl3 113 . 2
4 ax9v 1639 . . 3
5 dveeq2 1884 . . . . . 6
6 ax9v 1639 . . . . . . 7
7 hba1 1723 . . . . . . . 8
8 sp 1719 . . . . . . . . . 10
9 equequ2 1652 . . . . . . . . . 10
108, 9syl 17 . . . . . . . . 9
1110notbid 287 . . . . . . . 8
127, 11albidh 1579 . . . . . . 7
136, 12mtbii 295 . . . . . 6
145, 13syl6com 33 . . . . 5
1514con3i 129 . . . 4
1615alrimiv 1619 . . 3
174, 16mt3 173 . 2
183, 17pm2.61i 158 1
 Colors of variables: wff set class Syntax hints:   wn 5   wi 6   wb 178  wal 1529 This theorem is referenced by:  ax9o  1894  a9e  1895  ax4567to4  27003  ax12-2  28372  ax12-4  28375 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870 This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1531
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