ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  drnf1 Unicode version

Theorem drnf1 1668
Description: Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)
Hypothesis
Ref Expression
drex2.1  |-  ( A. x  x  =  y  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
drnf1  |-  ( A. x  x  =  y  ->  ( F/ x ph  <->  F/ y ps ) )

Proof of Theorem drnf1
StepHypRef Expression
1 drex2.1 . . . 4  |-  ( A. x  x  =  y  ->  ( ph  <->  ps )
)
21dral1 1665 . . . 4  |-  ( A. x  x  =  y  ->  ( A. x ph  <->  A. y ps ) )
31, 2imbi12d 232 . . 3  |-  ( A. x  x  =  y  ->  ( ( ph  ->  A. x ph )  <->  ( ps  ->  A. y ps )
) )
43dral1 1665 . 2  |-  ( A. x  x  =  y  ->  ( A. x (
ph  ->  A. x ph )  <->  A. y ( ps  ->  A. y ps ) ) )
5 df-nf 1395 . 2  |-  ( F/ x ph  <->  A. x
( ph  ->  A. x ph ) )
6 df-nf 1395 . 2  |-  ( F/ y ps  <->  A. y
( ps  ->  A. y ps ) )
74, 5, 63bitr4g 221 1  |-  ( A. x  x  =  y  ->  ( F/ x ph  <->  F/ y ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1287   F/wnf 1394
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-nf 1395
This theorem is referenced by:  drnfc1  2245
  Copyright terms: Public domain W3C validator