Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  alrim3con13v Structured version   Unicode version

Theorem alrim3con13v 28617
Description: Closed form of alrimi 1781 with 2 additional conjuncts having no occurences of the quantifying variable. This proof is 19.21a3con13vVD 28964 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
alrim3con13v  |-  ( (
ph  ->  A. x ph )  ->  ( ( ps  /\  ph 
/\  ch )  ->  A. x
( ps  /\  ph  /\ 
ch ) ) )
Distinct variable groups:    ps, x    ch, x
Allowed substitution hint:    ph( x)

Proof of Theorem alrim3con13v
StepHypRef Expression
1 simp1 957 . . . . 5  |-  ( ( ps  /\  ph  /\  ch )  ->  ps )
21a1i 11 . . . 4  |-  ( (
ph  ->  A. x ph )  ->  ( ( ps  /\  ph 
/\  ch )  ->  ps ) )
3 ax-17 1626 . . . 4  |-  ( ps 
->  A. x ps )
42, 3syl6 31 . . 3  |-  ( (
ph  ->  A. x ph )  ->  ( ( ps  /\  ph 
/\  ch )  ->  A. x ps ) )
5 simp2 958 . . . 4  |-  ( ( ps  /\  ph  /\  ch )  ->  ph )
65imim1i 56 . . 3  |-  ( (
ph  ->  A. x ph )  ->  ( ( ps  /\  ph 
/\  ch )  ->  A. x ph ) )
7 simp3 959 . . . . 5  |-  ( ( ps  /\  ph  /\  ch )  ->  ch )
87a1i 11 . . . 4  |-  ( (
ph  ->  A. x ph )  ->  ( ( ps  /\  ph 
/\  ch )  ->  ch ) )
9 ax-17 1626 . . . 4  |-  ( ch 
->  A. x ch )
108, 9syl6 31 . . 3  |-  ( (
ph  ->  A. x ph )  ->  ( ( ps  /\  ph 
/\  ch )  ->  A. x ch ) )
114, 6, 103jcad 1135 . 2  |-  ( (
ph  ->  A. x ph )  ->  ( ( ps  /\  ph 
/\  ch )  ->  ( A. x ps  /\  A. x ph  /\  A. x ch ) ) )
12 19.26-3an 1605 . 2  |-  ( A. x ( ps  /\  ph 
/\  ch )  <->  ( A. x ps  /\  A. x ph  /\  A. x ch ) )
1311, 12syl6ibr 219 1  |-  ( (
ph  ->  A. x ph )  ->  ( ( ps  /\  ph 
/\  ch )  ->  A. x
( ps  /\  ph  /\ 
ch ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936   A.wal 1549
This theorem is referenced by:  tratrbVD  28973
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938
  Copyright terms: Public domain W3C validator