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Theorem a9e2ndeq 28624
 Description: "At least two sets exist" expressed in the form of dtru 4217 is logically equivalent to the same expressed in a form similar to a9e 1904 if dtru 4217 is false implies . a9e2ndeq 28624 is derived from a9e2ndeqVD 29001. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
a9e2ndeq
Distinct variable groups:   ,   ,   ,   ,

Proof of Theorem a9e2ndeq
StepHypRef Expression
1 a9e2nd 28623 . . 3
2 a9e2eq 28622 . . . 4
31a1d 22 . . . 4
42, 3pm2.61i 156 . . 3
51, 4jaoi 368 . 2
6 olc 373 . . . 4
76a1d 22 . . 3
8 excom 1798 . . . . . 6
9 neeq1 2467 . . . . . . . . . . . . 13
109biimprcd 216 . . . . . . . . . . . 12
1110adantrd 454 . . . . . . . . . . 11
12 simpr 447 . . . . . . . . . . . 12
1312a1i 10 . . . . . . . . . . 11
14 neeq2 2468 . . . . . . . . . . . 12
1514biimprcd 216 . . . . . . . . . . 11
1611, 13, 15ee22 1352 . . . . . . . . . 10
17 sp 1728 . . . . . . . . . . 11
1817necon3ai 2499 . . . . . . . . . 10
1916, 18syl6 29 . . . . . . . . 9
2019eximdv 1612 . . . . . . . 8
21 nfnae 1909 . . . . . . . . 9
222119.9 1795 . . . . . . . 8
2320, 22syl6ib 217 . . . . . . 7
2423eximdv 1612 . . . . . 6
258, 24syl5bi 208 . . . . 5
26 nfnae 1909 . . . . . 6
272619.9 1795 . . . . 5
2825, 27syl6ib 217 . . . 4
29 orc 374 . . . 4
3028, 29syl6 29 . . 3
317, 30pm2.61ine 2535 . 2
325, 31impbii 180 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 176   wo 357   wa 358  wal 1530  wex 1531   wceq 1632   wne 2459 This theorem is referenced by:  2sb5nd  28625  2uasbanh  28626  2sb5ndVD  29002  2uasbanhVD  29003  2sb5ndALT  29025 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-ne 2461  df-v 2803
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