ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  19.26-3an Unicode version

Theorem 19.26-3an 1413
Description: Theorem 19.26 of [Margaris] p. 90 with triple conjunction. (Contributed by NM, 13-Sep-2011.)
Assertion
Ref Expression
19.26-3an  |-  ( A. x ( ph  /\  ps  /\  ch )  <->  ( A. x ph  /\  A. x ps  /\  A. x ch ) )

Proof of Theorem 19.26-3an
StepHypRef Expression
1 19.26 1411 . . 3  |-  ( A. x ( ( ph  /\ 
ps )  /\  ch ) 
<->  ( A. x (
ph  /\  ps )  /\  A. x ch )
)
2 19.26 1411 . . . 4  |-  ( A. x ( ph  /\  ps )  <->  ( A. x ph  /\  A. x ps ) )
32anbi1i 446 . . 3  |-  ( ( A. x ( ph  /\ 
ps )  /\  A. x ch )  <->  ( ( A. x ph  /\  A. x ps )  /\  A. x ch ) )
41, 3bitri 182 . 2  |-  ( A. x ( ( ph  /\ 
ps )  /\  ch ) 
<->  ( ( A. x ph  /\  A. x ps )  /\  A. x ch ) )
5 df-3an 922 . . 3  |-  ( (
ph  /\  ps  /\  ch ) 
<->  ( ( ph  /\  ps )  /\  ch )
)
65albii 1400 . 2  |-  ( A. x ( ph  /\  ps  /\  ch )  <->  A. x
( ( ph  /\  ps )  /\  ch )
)
7 df-3an 922 . 2  |-  ( ( A. x ph  /\  A. x ps  /\  A. x ch )  <->  ( ( A. x ph  /\  A. x ps )  /\  A. x ch ) )
84, 6, 73bitr4i 210 1  |-  ( A. x ( ph  /\  ps  /\  ch )  <->  ( A. x ph  /\  A. x ps  /\  A. x ch ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    /\ w3a 920   A.wal 1283
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-5 1377  ax-gen 1379
This theorem depends on definitions:  df-bi 115  df-3an 922
This theorem is referenced by:  hb3and  1420
  Copyright terms: Public domain W3C validator