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Theorem alrim3con13v 31556
Description: Closed form of alrimi 1816 with 2 additional conjuncts having no occurrences of the quantifying variable. This proof is 19.21a3con13vVD 31905 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 988 . . . . 5  |-  ( ( ps  /\  ph  /\  ch )  ->  ps )
21a1i 11 . . . 4  |-  ( (
ph  ->  A. x ph )  ->  ( ( ps  /\  ph 
/\  ch )  ->  ps ) )
3 ax-5 1671 . . . 4  |-  ( ps 
->  A. x ps )
42, 3syl6 33 . . 3  |-  ( (
ph  ->  A. x ph )  ->  ( ( ps  /\  ph 
/\  ch )  ->  A. x ps ) )
5 simp2 989 . . . 4  |-  ( ( ps  /\  ph  /\  ch )  ->  ph )
65imim1i 58 . . 3  |-  ( (
ph  ->  A. x ph )  ->  ( ( ps  /\  ph 
/\  ch )  ->  A. x ph ) )
7 simp3 990 . . . . 5  |-  ( ( ps  /\  ph  /\  ch )  ->  ch )
87a1i 11 . . . 4  |-  ( (
ph  ->  A. x ph )  ->  ( ( ps  /\  ph 
/\  ch )  ->  ch ) )
9 ax-5 1671 . . . 4  |-  ( ch 
->  A. x ch )
108, 9syl6 33 . . 3  |-  ( (
ph  ->  A. x ph )  ->  ( ( ps  /\  ph 
/\  ch )  ->  A. x ch ) )
114, 6, 103jcad 1169 . 2  |-  ( (
ph  ->  A. x ph )  ->  ( ( ps  /\  ph 
/\  ch )  ->  ( A. x ps  /\  A. x ph  /\  A. x ch ) ) )
12 19.26-3an 1650 . 2  |-  ( A. x ( ps  /\  ph 
/\  ch )  <->  ( A. x ps  /\  A. x ph  /\  A. x ch ) )
1311, 12syl6ibr 227 1  |-  ( (
ph  ->  A. x ph )  ->  ( ( ps  /\  ph 
/\  ch )  ->  A. x
( ps  /\  ph  /\ 
ch ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965   A.wal 1368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 967
This theorem is referenced by:  tratrbVD  31914
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