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Theorem mdbr3 25700
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 25697 . 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 24901 . . . . . . . 8  |-  ( ( x  e.  CH  /\  B  e.  CH )  ->  ( x  i^i  B
)  e.  CH )
3 inss2 3570 . . . . . . . . 9  |-  ( x  i^i  B )  C_  B
4 sseq1 3376 . . . . . . . . . . 11  |-  ( y  =  ( x  i^i 
B )  ->  (
y  C_  B  <->  ( x  i^i  B )  C_  B
) )
5 oveq1 6097 . . . . . . . . . . . . 13  |-  ( y  =  ( x  i^i 
B )  ->  (
y  vH  A )  =  ( ( x  i^i  B )  vH  A ) )
65ineq1d 3550 . . . . . . . . . . . 12  |-  ( y  =  ( x  i^i 
B )  ->  (
( y  vH  A
)  i^i  B )  =  ( ( ( x  i^i  B )  vH  A )  i^i 
B ) )
7 oveq1 6097 . . . . . . . . . . . 12  |-  ( y  =  ( x  i^i 
B )  ->  (
y  vH  ( A  i^i  B ) )  =  ( ( x  i^i 
B )  vH  ( A  i^i  B ) ) )
86, 7eqeq12d 2456 . . . . . . . . . . 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 320 . . . . . . . . . 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 3068 . . . . . . . . 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 43 . . . . . . . 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 16 . . . . . . 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 434 . . . . . 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 81 . . . . 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 2804 . . . 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 3342 . . . . . . . . . . 11  |-  ( x 
C_  B  <->  x  =  ( x  i^i  B ) )
1716biimpi 194 . . . . . . . . . 10  |-  ( x 
C_  B  ->  x  =  ( x  i^i 
B ) )
1817oveq1d 6105 . . . . . . . . 9  |-  ( x 
C_  B  ->  (
x  vH  A )  =  ( ( x  i^i  B )  vH  A ) )
1918ineq1d 3550 . . . . . . . 8  |-  ( x 
C_  B  ->  (
( x  vH  A
)  i^i  B )  =  ( ( ( x  i^i  B )  vH  A )  i^i 
B ) )
2017oveq1d 6105 . . . . . . . 8  |-  ( x 
C_  B  ->  (
x  vH  ( A  i^i  B ) )  =  ( ( x  i^i 
B )  vH  ( A  i^i  B ) ) )
2119, 20eqeq12d 2456 . . . . . . 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 225 . . . . . 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 2790 . . . . 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 3376 . . . . . . 7  |-  ( x  =  y  ->  (
x  C_  B  <->  y  C_  B ) )
25 oveq1 6097 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  vH  A )  =  ( y  vH  A ) )
2625ineq1d 3550 . . . . . . . 8  |-  ( x  =  y  ->  (
( x  vH  A
)  i^i  B )  =  ( ( y  vH  A )  i^i 
B ) )
27 oveq1 6097 . . . . . . . 8  |-  ( x  =  y  ->  (
x  vH  ( A  i^i  B ) )  =  ( y  vH  ( A  i^i  B ) ) )
2826, 27eqeq12d 2456 . . . . . . 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 320 . . . . . 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 2946 . . . . 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 196 . . . 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 203 . . 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 466 . 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 253 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2714    i^i cin 3326    C_ wss 3327   class class class wbr 4291  (class class class)co 6090   CHcch 24330    vH chj 24334    MH cmd 24367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-i2m1 9349  ax-1ne0 9350  ax-rrecex 9353  ax-cnre 9354  ax-hilex 24400  ax-hfvadd 24401  ax-hv0cl 24404  ax-hfvmul 24406
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-recs 6831  df-rdg 6865  df-map 7215  df-nn 10322  df-hlim 24373  df-sh 24608  df-ch 24623  df-md 25683
This theorem is referenced by: (None)
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