ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ax11b Structured version   GIF version

Theorem ax11b 1704
Description: A bidirectional version of ax-11o 1701. (Contributed by NM, 30-Jun-2006.)
Assertion
Ref Expression
ax11b ((¬ x x = y x = y) → (φx(x = yφ)))

Proof of Theorem ax11b
StepHypRef Expression
1 ax11o 1700 . . 3 x x = y → (x = y → (φx(x = yφ))))
21imp 115 . 2 ((¬ x x = y x = y) → (φx(x = yφ)))
3 ax-4 1397 . . . 4 (x(x = yφ) → (x = yφ))
43com12 27 . . 3 (x = y → (x(x = yφ) → φ))
54adantl 262 . 2 ((¬ x x = y x = y) → (x(x = yφ) → φ))
62, 5impbid 120 1 ((¬ x x = y x = y) → (φx(x = yφ)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98  wal 1240
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-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator