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Theorem rnlem 931
Description: Lemma used in construction of real numbers. (Contributed by NM, 4-Sep-1995.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
rnlem  |-  ( ( ( ph  /\  ps )  /\  ( ch  /\  th ) )  <->  ( (
( ph  /\  ch )  /\  ( ps  /\  th ) )  /\  (
( ph  /\  th )  /\  ( ps  /\  ch ) ) ) )

Proof of Theorem rnlem
StepHypRef Expression
1 an4 797 . . . 4  |-  ( ( ( ph  /\  ps )  /\  ( ch  /\  th ) )  <->  ( ( ph  /\  ch )  /\  ( ps  /\  th )
) )
21biimpi 186 . . 3  |-  ( ( ( ph  /\  ps )  /\  ( ch  /\  th ) )  ->  (
( ph  /\  ch )  /\  ( ps  /\  th ) ) )
3 an42 798 . . . 4  |-  ( ( ( ph  /\  th )  /\  ( ps  /\  ch ) )  <->  ( ( ph  /\  ps )  /\  ( ch  /\  th )
) )
43biimpri 197 . . 3  |-  ( ( ( ph  /\  ps )  /\  ( ch  /\  th ) )  ->  (
( ph  /\  th )  /\  ( ps  /\  ch ) ) )
52, 4jca 518 . 2  |-  ( ( ( ph  /\  ps )  /\  ( ch  /\  th ) )  ->  (
( ( ph  /\  ch )  /\  ( ps  /\  th ) )  /\  ( ( ph  /\ 
th )  /\  ( ps  /\  ch ) ) ) )
63biimpi 186 . . 3  |-  ( ( ( ph  /\  th )  /\  ( ps  /\  ch ) )  ->  (
( ph  /\  ps )  /\  ( ch  /\  th ) ) )
76adantl 452 . 2  |-  ( ( ( ( ph  /\  ch )  /\  ( ps  /\  th ) )  /\  ( ( ph  /\ 
th )  /\  ( ps  /\  ch ) ) )  ->  ( ( ph  /\  ps )  /\  ( ch  /\  th )
) )
85, 7impbii 180 1  |-  ( ( ( ph  /\  ps )  /\  ( ch  /\  th ) )  <->  ( (
( ph  /\  ch )  /\  ( ps  /\  th ) )  /\  (
( ph  /\  th )  /\  ( ps  /\  ch ) ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358
This theorem is referenced by:  mulcmpblnr  8712
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-an 360
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