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Theorem meetval2lem 15526
Description: Lemma for meetval2 15527 and meeteu 15528. (Contributed by NM, 12-Sep-2018.) TODO: combine this through meeteu into meetlem?
Hypotheses
Ref Expression
meetval2.b  |-  B  =  ( Base `  K
)
meetval2.l  |-  .<_  =  ( le `  K )
meetval2.m  |-  ./\  =  ( meet `  K )
meetval2.k  |-  ( ph  ->  K  e.  V )
meetval2.x  |-  ( ph  ->  X  e.  B )
meetval2.y  |-  ( ph  ->  Y  e.  B )
Assertion
Ref Expression
meetval2lem  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( ( A. y  e.  { X ,  Y } x  .<_  y  /\  A. z  e.  B  ( A. y  e.  { X ,  Y }
z  .<_  y  ->  z  .<_  x ) )  <->  ( (
x  .<_  X  /\  x  .<_  Y )  /\  A. z  e.  B  (
( z  .<_  X  /\  z  .<_  Y )  -> 
z  .<_  x ) ) ) )
Distinct variable groups:    x, z, B    x,  ./\ , z    x, y, K, z    y,  .<_    x, X, y, z    x, Y, y, z
Allowed substitution hints:    ph( x, y, z)    B( y)    .<_ ( x, z)    ./\ ( y)    V( x, y, z)

Proof of Theorem meetval2lem
StepHypRef Expression
1 breq2 4441 . . 3  |-  ( y  =  X  ->  (
x  .<_  y  <->  x  .<_  X ) )
2 breq2 4441 . . 3  |-  ( y  =  Y  ->  (
x  .<_  y  <->  x  .<_  Y ) )
31, 2ralprg 4063 . 2  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( A. y  e. 
{ X ,  Y } x  .<_  y  <->  ( x  .<_  X  /\  x  .<_  Y ) ) )
4 breq2 4441 . . . . 5  |-  ( y  =  X  ->  (
z  .<_  y  <->  z  .<_  X ) )
5 breq2 4441 . . . . 5  |-  ( y  =  Y  ->  (
z  .<_  y  <->  z  .<_  Y ) )
64, 5ralprg 4063 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( A. y  e. 
{ X ,  Y } z  .<_  y  <->  ( z  .<_  X  /\  z  .<_  Y ) ) )
76imbi1d 317 . . 3  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( ( A. y  e.  { X ,  Y } z  .<_  y  -> 
z  .<_  x )  <->  ( (
z  .<_  X  /\  z  .<_  Y )  ->  z  .<_  x ) ) )
87ralbidv 2882 . 2  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( A. z  e.  B  ( A. y  e.  { X ,  Y } z  .<_  y  -> 
z  .<_  x )  <->  A. z  e.  B  ( (
z  .<_  X  /\  z  .<_  Y )  ->  z  .<_  x ) ) )
93, 8anbi12d 710 1  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( ( A. y  e.  { X ,  Y } x  .<_  y  /\  A. z  e.  B  ( A. y  e.  { X ,  Y }
z  .<_  y  ->  z  .<_  x ) )  <->  ( (
x  .<_  X  /\  x  .<_  Y )  /\  A. z  e.  B  (
( z  .<_  X  /\  z  .<_  Y )  -> 
z  .<_  x ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804   A.wral 2793   {cpr 4016   class class class wbr 4437   ` cfv 5578   Basecbs 14509   lecple 14581   meetcmee 15448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ral 2798  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438
This theorem is referenced by:  meetval2  15527  meeteu  15528
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