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Theorem meetval2lem 15498
Description: Lemma for meetval2 15499 and meeteu 15500. (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 4444 . . 3  |-  ( y  =  X  ->  (
x  .<_  y  <->  x  .<_  X ) )
2 breq2 4444 . . 3  |-  ( y  =  Y  ->  (
x  .<_  y  <->  x  .<_  Y ) )
31, 2ralprg 4069 . 2  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( A. y  e. 
{ X ,  Y } x  .<_  y  <->  ( x  .<_  X  /\  x  .<_  Y ) ) )
4 breq2 4444 . . . . 5  |-  ( y  =  X  ->  (
z  .<_  y  <->  z  .<_  X ) )
5 breq2 4444 . . . . 5  |-  ( y  =  Y  ->  (
z  .<_  y  <->  z  .<_  Y ) )
64, 5ralprg 4069 . . . 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 2896 . 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 1374    e. wcel 1762   A.wral 2807   {cpr 4022   class class class wbr 4440   ` cfv 5579   Basecbs 14479   lecple 14551   meetcmee 15421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ral 2812  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441
This theorem is referenced by:  meetval2  15499  meeteu  15500
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