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

Theorem bm1.3ii 4144
 Description: Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 4141. 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 1689 . . . . . . . 8
32imbi2d 307 . . . . . . 7
43albidv 1611 . . . . . 6
54cbvexv 1943 . . . . 5
61, 5mpbi 199 . . . 4
7 ax-sep 4141 . . . 4
86, 7pm3.2i 441 . . 3
98exan 1823 . 2
10 19.42v 1846 . . . 4
11 bimsc1 904 . . . . . 6
1211alanimi 1549 . . . . 5
1312eximi 1563 . . . 4
1410, 13sylbir 204 . . 3
1514exlimiv 1666 . 2
169, 15ax-mp 8 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358  wal 1527  wex 1528   wceq 1623   wcel 1684 This theorem is referenced by:  axpow3  4191  pwex  4193  zfpair2  4215  axun2  4514  uniex2  4515 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-sep 4141 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532
 Copyright terms: Public domain W3C validator