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Theorem mdbr3 27629
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 27626 . 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 26831 . . . . . . . 8  |-  ( ( x  e.  CH  /\  B  e.  CH )  ->  ( x  i^i  B
)  e.  CH )
3 inss2 3660 . . . . . . . . 9  |-  ( x  i^i  B )  C_  B
4 sseq1 3463 . . . . . . . . . . 11  |-  ( y  =  ( x  i^i 
B )  ->  (
y  C_  B  <->  ( x  i^i  B )  C_  B
) )
5 oveq1 6285 . . . . . . . . . . . . 13  |-  ( y  =  ( x  i^i 
B )  ->  (
y  vH  A )  =  ( ( x  i^i  B )  vH  A ) )
65ineq1d 3640 . . . . . . . . . . . 12  |-  ( y  =  ( x  i^i 
B )  ->  (
( y  vH  A
)  i^i  B )  =  ( ( ( x  i^i  B )  vH  A )  i^i 
B ) )
7 oveq1 6285 . . . . . . . . . . . 12  |-  ( y  =  ( x  i^i 
B )  ->  (
y  vH  ( A  i^i  B ) )  =  ( ( x  i^i 
B )  vH  ( A  i^i  B ) ) )
86, 7eqeq12d 2424 . . . . . . . . . . 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 318 . . . . . . . . . 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 3156 . . . . . . . . 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 41 . . . . . . . 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 17 . . . . . . 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 432 . . . . . 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 2820 . . . 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 3429 . . . . . . . . . . 11  |-  ( x 
C_  B  <->  x  =  ( x  i^i  B ) )
1716biimpi 194 . . . . . . . . . 10  |-  ( x 
C_  B  ->  x  =  ( x  i^i 
B ) )
1817oveq1d 6293 . . . . . . . . 9  |-  ( x 
C_  B  ->  (
x  vH  A )  =  ( ( x  i^i  B )  vH  A ) )
1918ineq1d 3640 . . . . . . . 8  |-  ( x 
C_  B  ->  (
( x  vH  A
)  i^i  B )  =  ( ( ( x  i^i  B )  vH  A )  i^i 
B ) )
2017oveq1d 6293 . . . . . . . 8  |-  ( x 
C_  B  ->  (
x  vH  ( A  i^i  B ) )  =  ( ( x  i^i 
B )  vH  ( A  i^i  B ) ) )
2119, 20eqeq12d 2424 . . . . . . 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 2797 . . . . 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 3463 . . . . . . 7  |-  ( x  =  y  ->  (
x  C_  B  <->  y  C_  B ) )
25 oveq1 6285 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  vH  A )  =  ( y  vH  A ) )
2625ineq1d 3640 . . . . . . . 8  |-  ( x  =  y  ->  (
( x  vH  A
)  i^i  B )  =  ( ( y  vH  A )  i^i 
B ) )
27 oveq1 6285 . . . . . . . 8  |-  ( x  =  y  ->  (
x  vH  ( A  i^i  B ) )  =  ( y  vH  ( A  i^i  B ) ) )
2826, 27eqeq12d 2424 . . . . . . 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 318 . . . . . 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 3034 . . . . 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 464 . 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 367    = wceq 1405    e. wcel 1842   A.wral 2754    i^i cin 3413    C_ wss 3414   class class class wbr 4395  (class class class)co 6278   CHcch 26260    vH chj 26264    MH cmd 26297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-i2m1 9590  ax-1ne0 9591  ax-rrecex 9594  ax-cnre 9595  ax-hilex 26330  ax-hfvadd 26331  ax-hv0cl 26334  ax-hfvmul 26336
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-map 7459  df-nn 10577  df-hlim 26303  df-sh 26538  df-ch 26553  df-md 27612
This theorem is referenced by: (None)
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