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Theorem mdbr3 22877
Description: Binary relation expressing the modular pair property. This version quantifies an equality instead of an inference. (Contributed by NM, 6-Jul-2004.) (New usage is discouraged.)
Assertion
Ref Expression
mdbr3  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH  B  <->  A. x  e.  CH  (
( ( x  i^i 
B )  vH  A
)  i^i  B )  =  ( ( x  i^i  B )  vH  ( A  i^i  B ) ) ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem mdbr3
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 mdbr 22874 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH  B  <->  A. y  e.  CH  (
y  C_  B  ->  ( ( y  vH  A
)  i^i  B )  =  ( y  vH  ( A  i^i  B ) ) ) ) )
2 chincl 22078 . . . . . . . 8  |-  ( ( x  e.  CH  /\  B  e.  CH )  ->  ( x  i^i  B
)  e.  CH )
3 inss2 3390 . . . . . . . . 9  |-  ( x  i^i  B )  C_  B
4 sseq1 3199 . . . . . . . . . . 11  |-  ( y  =  ( x  i^i 
B )  ->  (
y  C_  B  <->  ( x  i^i  B )  C_  B
) )
5 oveq1 5865 . . . . . . . . . . . . 13  |-  ( y  =  ( x  i^i 
B )  ->  (
y  vH  A )  =  ( ( x  i^i  B )  vH  A ) )
65ineq1d 3369 . . . . . . . . . . . 12  |-  ( y  =  ( x  i^i 
B )  ->  (
( y  vH  A
)  i^i  B )  =  ( ( ( x  i^i  B )  vH  A )  i^i 
B ) )
7 oveq1 5865 . . . . . . . . . . . 12  |-  ( y  =  ( x  i^i 
B )  ->  (
y  vH  ( A  i^i  B ) )  =  ( ( x  i^i 
B )  vH  ( A  i^i  B ) ) )
86, 7eqeq12d 2297 . . . . . . . . . . 11  |-  ( y  =  ( x  i^i 
B )  ->  (
( ( y  vH  A )  i^i  B
)  =  ( y  vH  ( A  i^i  B ) )  <->  ( (
( x  i^i  B
)  vH  A )  i^i  B )  =  ( ( x  i^i  B
)  vH  ( A  i^i  B ) ) ) )
94, 8imbi12d 311 . . . . . . . . . 10  |-  ( y  =  ( x  i^i 
B )  ->  (
( y  C_  B  ->  ( ( y  vH  A )  i^i  B
)  =  ( y  vH  ( A  i^i  B ) ) )  <->  ( (
x  i^i  B )  C_  B  ->  ( (
( x  i^i  B
)  vH  A )  i^i  B )  =  ( ( x  i^i  B
)  vH  ( A  i^i  B ) ) ) ) )
109rspcv 2880 . . . . . . . . 9  |-  ( ( x  i^i  B )  e.  CH  ->  ( A. y  e.  CH  (
y  C_  B  ->  ( ( y  vH  A
)  i^i  B )  =  ( y  vH  ( A  i^i  B ) ) )  ->  (
( x  i^i  B
)  C_  B  ->  ( ( ( x  i^i 
B )  vH  A
)  i^i  B )  =  ( ( x  i^i  B )  vH  ( A  i^i  B ) ) ) ) )
113, 10mpii 39 . . . . . . . 8  |-  ( ( x  i^i  B )  e.  CH  ->  ( A. y  e.  CH  (
y  C_  B  ->  ( ( y  vH  A
)  i^i  B )  =  ( y  vH  ( A  i^i  B ) ) )  ->  (
( ( x  i^i 
B )  vH  A
)  i^i  B )  =  ( ( x  i^i  B )  vH  ( A  i^i  B ) ) ) )
122, 11syl 15 . . . . . . 7  |-  ( ( x  e.  CH  /\  B  e.  CH )  ->  ( A. y  e. 
CH  ( y  C_  B  ->  ( ( y  vH  A )  i^i 
B )  =  ( y  vH  ( A  i^i  B ) ) )  ->  ( (
( x  i^i  B
)  vH  A )  i^i  B )  =  ( ( x  i^i  B
)  vH  ( A  i^i  B ) ) ) )
1312ex 423 . . . . . 6  |-  ( x  e.  CH  ->  ( B  e.  CH  ->  ( A. y  e.  CH  (
y  C_  B  ->  ( ( y  vH  A
)  i^i  B )  =  ( y  vH  ( A  i^i  B ) ) )  ->  (
( ( x  i^i 
B )  vH  A
)  i^i  B )  =  ( ( x  i^i  B )  vH  ( A  i^i  B ) ) ) ) )
1413com3l 75 . . . . 5  |-  ( B  e.  CH  ->  ( A. y  e.  CH  (
y  C_  B  ->  ( ( y  vH  A
)  i^i  B )  =  ( y  vH  ( A  i^i  B ) ) )  ->  (
x  e.  CH  ->  ( ( ( x  i^i 
B )  vH  A
)  i^i  B )  =  ( ( x  i^i  B )  vH  ( A  i^i  B ) ) ) ) )
1514ralrimdv 2632 . . . 4  |-  ( B  e.  CH  ->  ( A. y  e.  CH  (
y  C_  B  ->  ( ( y  vH  A
)  i^i  B )  =  ( y  vH  ( A  i^i  B ) ) )  ->  A. x  e.  CH  ( ( ( x  i^i  B )  vH  A )  i^i 
B )  =  ( ( x  i^i  B
)  vH  ( A  i^i  B ) ) ) )
16 dfss 3167 . . . . . . . . . . 11  |-  ( x 
C_  B  <->  x  =  ( x  i^i  B ) )
1716biimpi 186 . . . . . . . . . 10  |-  ( x 
C_  B  ->  x  =  ( x  i^i 
B ) )
1817oveq1d 5873 . . . . . . . . 9  |-  ( x 
C_  B  ->  (
x  vH  A )  =  ( ( x  i^i  B )  vH  A ) )
1918ineq1d 3369 . . . . . . . 8  |-  ( x 
C_  B  ->  (
( x  vH  A
)  i^i  B )  =  ( ( ( x  i^i  B )  vH  A )  i^i 
B ) )
2017oveq1d 5873 . . . . . . . 8  |-  ( x 
C_  B  ->  (
x  vH  ( A  i^i  B ) )  =  ( ( x  i^i 
B )  vH  ( A  i^i  B ) ) )
2119, 20eqeq12d 2297 . . . . . . 7  |-  ( x 
C_  B  ->  (
( ( x  vH  A )  i^i  B
)  =  ( x  vH  ( A  i^i  B ) )  <->  ( (
( x  i^i  B
)  vH  A )  i^i  B )  =  ( ( x  i^i  B
)  vH  ( A  i^i  B ) ) ) )
2221biimprcd 216 . . . . . 6  |-  ( ( ( ( x  i^i 
B )  vH  A
)  i^i  B )  =  ( ( x  i^i  B )  vH  ( A  i^i  B ) )  ->  ( x  C_  B  ->  ( (
x  vH  A )  i^i  B )  =  ( x  vH  ( A  i^i  B ) ) ) )
2322ralimi 2618 . . . . 5  |-  ( A. x  e.  CH  ( ( ( x  i^i  B
)  vH  A )  i^i  B )  =  ( ( x  i^i  B
)  vH  ( A  i^i  B ) )  ->  A. x  e.  CH  (
x  C_  B  ->  ( ( x  vH  A
)  i^i  B )  =  ( x  vH  ( A  i^i  B ) ) ) )
24 sseq1 3199 . . . . . . 7  |-  ( x  =  y  ->  (
x  C_  B  <->  y  C_  B ) )
25 oveq1 5865 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  vH  A )  =  ( y  vH  A ) )
2625ineq1d 3369 . . . . . . . 8  |-  ( x  =  y  ->  (
( x  vH  A
)  i^i  B )  =  ( ( y  vH  A )  i^i 
B ) )
27 oveq1 5865 . . . . . . . 8  |-  ( x  =  y  ->  (
x  vH  ( A  i^i  B ) )  =  ( y  vH  ( A  i^i  B ) ) )
2826, 27eqeq12d 2297 . . . . . . 7  |-  ( x  =  y  ->  (
( ( x  vH  A )  i^i  B
)  =  ( x  vH  ( A  i^i  B ) )  <->  ( (
y  vH  A )  i^i  B )  =  ( y  vH  ( A  i^i  B ) ) ) )
2924, 28imbi12d 311 . . . . . 6  |-  ( x  =  y  ->  (
( x  C_  B  ->  ( ( x  vH  A )  i^i  B
)  =  ( x  vH  ( A  i^i  B ) ) )  <->  ( y  C_  B  ->  ( (
y  vH  A )  i^i  B )  =  ( y  vH  ( A  i^i  B ) ) ) ) )
3029cbvralv 2764 . . . . 5  |-  ( A. x  e.  CH  ( x 
C_  B  ->  (
( x  vH  A
)  i^i  B )  =  ( x  vH  ( A  i^i  B ) ) )  <->  A. y  e.  CH  ( y  C_  B  ->  ( ( y  vH  A )  i^i 
B )  =  ( y  vH  ( A  i^i  B ) ) ) )
3123, 30sylib 188 . . . 4  |-  ( A. x  e.  CH  ( ( ( x  i^i  B
)  vH  A )  i^i  B )  =  ( ( x  i^i  B
)  vH  ( A  i^i  B ) )  ->  A. y  e.  CH  (
y  C_  B  ->  ( ( y  vH  A
)  i^i  B )  =  ( y  vH  ( A  i^i  B ) ) ) )
3215, 31impbid1 194 . . 3  |-  ( B  e.  CH  ->  ( A. y  e.  CH  (
y  C_  B  ->  ( ( y  vH  A
)  i^i  B )  =  ( y  vH  ( A  i^i  B ) ) )  <->  A. x  e.  CH  ( ( ( x  i^i  B )  vH  A )  i^i 
B )  =  ( ( x  i^i  B
)  vH  ( A  i^i  B ) ) ) )
3332adantl 452 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A. y  e. 
CH  ( y  C_  B  ->  ( ( y  vH  A )  i^i 
B )  =  ( y  vH  ( A  i^i  B ) ) )  <->  A. x  e.  CH  ( ( ( x  i^i  B )  vH  A )  i^i  B
)  =  ( ( x  i^i  B )  vH  ( A  i^i  B ) ) ) )
341, 33bitrd 244 1  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH  B  <->  A. x  e.  CH  (
( ( x  i^i 
B )  vH  A
)  i^i  B )  =  ( ( x  i^i  B )  vH  ( A  i^i  B ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    i^i cin 3151    C_ wss 3152   class class class wbr 4023  (class class class)co 5858   CHcch 21509    vH chj 21513    MH cmd 21546
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-i2m1 8805  ax-1ne0 8806  ax-rrecex 8809  ax-cnre 8810  ax-hilex 21579  ax-hfvadd 21580  ax-hv0cl 21583  ax-hfvmul 21585
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-map 6774  df-nn 9747  df-hlim 21552  df-sh 21786  df-ch 21801  df-md 22860
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