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Theorem bm1.3ii 4059
 Description: Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 4056. 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 2117 . . . . . . . 8
32imbi2d 229 . . . . . . 7
43albidv 1792 . . . . . 6
54cbvexv 1886 . . . . 5
61, 5mpbi 144 . . . 4
7 ax-sep 4056 . . . 4
86, 7pm3.2i 270 . . 3
98exan 1669 . 2
10 19.42v 1874 . . . 4
11 bimsc1 948 . . . . . 6
1211alanimi 1436 . . . . 5
1312eximi 1576 . . . 4
1410, 13sylbir 134 . . 3
1514exlimiv 1574 . 2
169, 15ax-mp 5 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wb 104  wal 1330  wex 1469 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-14 2115  ax-sep 4056 This theorem depends on definitions:  df-bi 116 This theorem is referenced by:  axpow3  4111  vpwex  4113  zfpair2  4143  axun2  4368  uniex2  4369
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