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Theorem bm1.3ii 3906
 Description: Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 3903. Similar to Theorem 1.3ii of [BellMachover] p. 463. (Contributed by NM, 5-Aug-1993.)
Hypothesis
Ref Expression
bm1.3ii.1
Assertion
Ref Expression
bm1.3ii
Distinct variable groups:   ,   ,
Allowed substitution hint:   ()

Proof of Theorem bm1.3ii
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 bm1.3ii.1 . . . . 5
2 elequ2 1617 . . . . . . . 8
32imbi2d 223 . . . . . . 7
43albidv 1721 . . . . . 6
54cbvexv 1811 . . . . 5
61, 5mpbi 137 . . . 4
7 ax-sep 3903 . . . 4
86, 7pm3.2i 261 . . 3
98exan 1599 . 2
10 19.42v 1802 . . . 4
11 bimsc1 881 . . . . . 6
1211alanimi 1364 . . . . 5
1312eximi 1507 . . . 4
1410, 13sylbir 129 . . 3
1514exlimiv 1505 . 2
169, 15ax-mp 7 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 101   wb 102  wal 1257  wex 1397 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-sep 3903 This theorem depends on definitions:  df-bi 114 This theorem is referenced by:  axpow3  3958  pwex  3960  zfpair2  3973  axun2  4200  uniex2  4201
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