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Theorem axc11n-16 2268
 Description: This theorem shows that, given ax-c16 2223, we can derive a version of ax-c11n 2219. However, it is weaker than ax-c11n 2219 because it has a distinct variable requirement. (Contributed by Andrew Salmon, 27-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
axc11n-16
Distinct variable group:   ,

Proof of Theorem axc11n-16
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ax-c16 2223 . . . 4
21alrimiv 1719 . . 3
32axc4i-o 2229 . 2
4 equequ1 1798 . . . . . 6
54cbvalv 2023 . . . . . . 7
65a1i 11 . . . . . 6
74, 6imbi12d 320 . . . . 5
87albidv 1713 . . . 4
98cbvalv 2023 . . 3
109biimpi 194 . 2
11 nfa1-o 2245 . . . . . . 7
121119.23 1910 . . . . . 6
1312albii 1640 . . . . 5
14 ax6ev 1749 . . . . . . . 8
15 pm2.27 39 . . . . . . . 8
1614, 15ax-mp 5 . . . . . . 7
1716alimi 1633 . . . . . 6
18 equequ2 1799 . . . . . . . . 9
1918spv 2011 . . . . . . . 8
2019sps-o 2238 . . . . . . 7
2120alcoms 1843 . . . . . 6
2217, 21syl 16 . . . . 5
2313, 22sylbi 195 . . . 4
2423alcoms 1843 . . 3
2524axc4i-o 2229 . 2
263, 10, 253syl 20 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  A.wal 1393  E.wex 1612 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-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-c5 2214  ax-c4 2215  ax-c7 2216  ax-c16 2223 This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1613  df-nf 1617
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