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Theorem bm1.3ii 3869
Description: Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 3866. 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 1598 . . . . . . . 8
32imbi2d 219 . . . . . . 7
43albidv 1702 . . . . . 6
54cbvexv 1792 . . . . 5
61, 5mpbi 133 . . . 4
7 ax-sep 3866 . . . 4
86, 7pm3.2i 257 . . 3
98exan 1580 . 2
10 19.42v 1783 . . . 4
11 bimsc1 869 . . . . . 6
1211alanimi 1345 . . . . 5
1312eximi 1488 . . . 4
1410, 13sylbir 125 . . 3
1514exlimiv 1486 . 2
169, 15ax-mp 7 1
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98  wal 1240  wex 1378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-sep 3866
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  axpow3  3921  pwex  3923  zfpair2  3936  axun2  4138  uniex2  4139
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