ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  bm1.3ii Structured version   GIF version

Theorem bm1.3ii 3869
Description: Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 3866. Similar to Theorem 1.3ii of [BellMachover] p. 463. (Contributed by NM, 5-Aug-1993.)
Hypothesis
Ref Expression
bm1.3ii.1 xy(φy x)
Assertion
Ref Expression
bm1.3ii xy(y xφ)
Distinct variable groups:   φ,x   x,y
Allowed substitution hint:   φ(y)

Proof of Theorem bm1.3ii
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 bm1.3ii.1 . . . . 5 xy(φy x)
2 elequ2 1598 . . . . . . . 8 (x = z → (y xy z))
32imbi2d 219 . . . . . . 7 (x = z → ((φy x) ↔ (φy z)))
43albidv 1702 . . . . . 6 (x = z → (y(φy x) ↔ y(φy z)))
54cbvexv 1792 . . . . 5 (xy(φy x) ↔ zy(φy z))
61, 5mpbi 133 . . . 4 zy(φy z)
7 ax-sep 3866 . . . 4 xy(y x ↔ (y z φ))
86, 7pm3.2i 257 . . 3 (zy(φy z) xy(y x ↔ (y z φ)))
98exan 1580 . 2 z(y(φy z) xy(y x ↔ (y z φ)))
10 19.42v 1783 . . . 4 (x(y(φy z) y(y x ↔ (y z φ))) ↔ (y(φy z) xy(y x ↔ (y z φ))))
11 bimsc1 869 . . . . . 6 (((φy z) (y x ↔ (y z φ))) → (y xφ))
1211alanimi 1345 . . . . 5 ((y(φy z) y(y x ↔ (y z φ))) → y(y xφ))
1312eximi 1488 . . . 4 (x(y(φy z) y(y x ↔ (y z φ))) → xy(y xφ))
1410, 13sylbir 125 . . 3 ((y(φy z) xy(y x ↔ (y z φ))) → xy(y xφ))
1514exlimiv 1486 . 2 (z(y(φy z) xy(y x ↔ (y z φ))) → xy(y xφ))
169, 15ax-mp 7 1 xy(y xφ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240  wex 1378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-sep 3866
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  axpow3  3921  pwex  3923  zfpair2  3936  axun2  4138  uniex2  4139
  Copyright terms: Public domain W3C validator