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Theorem bm1.3ii 4576
 Description: Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 4573. Similar to Theorem 1.3ii of [BellMachover] p. 463. (Contributed by NM, 21-Jun-1993.)
Hypothesis
Ref Expression
bm1.3ii.1
Assertion
Ref Expression
bm1.3ii
Distinct variable groups:   ,   ,

Proof of Theorem bm1.3ii
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 bm1.3ii.1 . . . . 5
2 elequ2 1823 . . . . . . . 8
32imbi2d 316 . . . . . . 7
43albidv 1713 . . . . . 6
54cbvexv 2024 . . . . 5
61, 5mpbi 208 . . . 4
7 ax-sep 4573 . . . 4
86, 7pm3.2i 455 . . 3
98exan 1973 . 2
10 19.42v 1775 . . . 4
11 bimsc1 938 . . . . . 6
1211alanimi 1637 . . . . 5
1312eximi 1656 . . . 4
1410, 13sylbir 213 . . 3
1514exlimiv 1722 . 2
169, 15ax-mp 5 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  E.wex 1612 This theorem is referenced by:  axpow3  4633  pwex  4635  zfpair2  4692  axun2  6594  uniex2  6595 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-sep 4573 This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1613  df-nf 1617
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