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Theorem 2uasbanh 28746
 Description: Distribute the unabbreviated form of proper substitution in and out of a conjunction. 2uasbanh 28746 is derived from 2uasbanhVD 29121. (Contributed by Alan Sare, 31-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
2uasbanh.1
Assertion
Ref Expression
2uasbanh
Distinct variable groups:   ,   ,   ,   ,
Allowed substitution hints:   (,,,)   (,,,)   (,,,)

Proof of Theorem 2uasbanh
StepHypRef Expression
1 simpl 445 . . . . 5
2 simprl 734 . . . . 5
31, 2jca 520 . . . 4
432eximi 1587 . . 3
5 simprr 735 . . . . 5
61, 5jca 520 . . . 4
762eximi 1587 . . 3
84, 7jca 520 . 2
9 2uasbanh.1 . . 3
109simplbi 448 . . . . . 6
11 simpl 445 . . . . . . . . . 10
12112eximi 1587 . . . . . . . . 9
1310, 12syl 16 . . . . . . . 8
14 a9e2ndeq 28744 . . . . . . . 8
1513, 14sylibr 205 . . . . . . 7
16 2sb5nd 28745 . . . . . . 7
1715, 16syl 16 . . . . . 6
1810, 17mpbird 225 . . . . 5
199simprbi 452 . . . . . 6
20 2sb5nd 28745 . . . . . . 7
2115, 20syl 16 . . . . . 6
2219, 21mpbird 225 . . . . 5
23 sban 2144 . . . . . . 7
2423sbbii 1667 . . . . . 6
25 sban 2144 . . . . . 6
2624, 25bitri 242 . . . . 5
2718, 22, 26sylanbrc 647 . . . 4
28 2sb5nd 28745 . . . . 5
2915, 28syl 16 . . . 4
3027, 29mpbid 203 . . 3
319, 30sylbir 206 . 2
328, 31impbii 182 1
 Colors of variables: wff set class Syntax hints:   wn 3   wb 178   wo 359   wa 360  wal 1550  wex 1551  wsb 1659 This theorem is referenced by:  2uasban  28747 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-ne 2607  df-v 2964
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