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Theorem bm1.3ii 4145
 Description: Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 4142. 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:   ,   ,
Dummy variable is distinct from all other variables.
Allowed substitution hint:   ()

Proof of Theorem bm1.3ii
StepHypRef Expression
1 bm1.3ii.1 . . . . 5
2 elequ2 1690 . . . . . . . 8
32imbi2d 309 . . . . . . 7
43albidv 1612 . . . . . 6
54cbvexv 1948 . . . . 5
61, 5mpbi 201 . . . 4
7 ax-sep 4142 . . . 4
86, 7pm3.2i 443 . . 3
98exan 1824 . 2
10 19.42v 1847 . . . 4
11 bimsc1 906 . . . . . 6
1211alanimi 1550 . . . . 5
1312eximi 1564 . . . 4
1410, 13sylbir 206 . . 3
1514exlimiv 1667 . 2
169, 15ax-mp 10 1
 Colors of variables: wff set class Syntax hints:   wi 6   wb 178   wa 360  wal 1528  wex 1529   wceq 1624   wcel 1685 This theorem is referenced by:  axpow3  4190  pwex  4192  zfpair2  4214  axun2  4513  uniex2  4514 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-sep 4142 This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533
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