Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  bm1.3ii Structured version   Unicode version

Theorem bm1.3ii 4492
 Description: Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 4489. 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:   ,   ,
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 1877 . . . . . . . 8
32imbi2d 317 . . . . . . 7
43albidv 1761 . . . . . 6
54cbvexv 2089 . . . . 5
61, 5mpbi 211 . . . 4
7 ax-sep 4489 . . . 4
86, 7pm3.2i 456 . . 3
98exan 2030 . 2
10 19.42v 1827 . . . 4
11 bimsc1 946 . . . . . 6
1211alanimi 1682 . . . . 5
1312eximi 1701 . . . 4
1410, 13sylbir 216 . . 3
1514exlimiv 1770 . 2
169, 15ax-mp 5 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 187   wa 370  wal 1435  wex 1657 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-sep 4489 This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658  df-nf 1662 This theorem is referenced by:  axpow3  4548  pwex  4550  zfpair2  4604  axun2  6543  uniex2  6544
 Copyright terms: Public domain W3C validator