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Theorem mdbr3 22893
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 22890 . 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 22094 . . . . . . . 8  |-  ( ( x  e.  CH  /\  B  e.  CH )  ->  ( x  i^i  B
)  e.  CH )
3 inss2 3403 . . . . . . . . 9  |-  ( x  i^i  B )  C_  B
4 sseq1 3212 . . . . . . . . . . 11  |-  ( y  =  ( x  i^i 
B )  ->  (
y  C_  B  <->  ( x  i^i  B )  C_  B
) )
5 oveq1 5881 . . . . . . . . . . . . 13  |-  ( y  =  ( x  i^i 
B )  ->  (
y  vH  A )  =  ( ( x  i^i  B )  vH  A ) )
65ineq1d 3382 . . . . . . . . . . . 12  |-  ( y  =  ( x  i^i 
B )  ->  (
( y  vH  A
)  i^i  B )  =  ( ( ( x  i^i  B )  vH  A )  i^i 
B ) )
7 oveq1 5881 . . . . . . . . . . . 12  |-  ( y  =  ( x  i^i 
B )  ->  (
y  vH  ( A  i^i  B ) )  =  ( ( x  i^i 
B )  vH  ( A  i^i  B ) ) )
86, 7eqeq12d 2310 . . . . . . . . . . 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 2893 . . . . . . . . 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 2645 . . . 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 3180 . . . . . . . . . . 11  |-  ( x 
C_  B  <->  x  =  ( x  i^i  B ) )
1716biimpi 186 . . . . . . . . . 10  |-  ( x 
C_  B  ->  x  =  ( x  i^i 
B ) )
1817oveq1d 5889 . . . . . . . . 9  |-  ( x 
C_  B  ->  (
x  vH  A )  =  ( ( x  i^i  B )  vH  A ) )
1918ineq1d 3382 . . . . . . . 8  |-  ( x 
C_  B  ->  (
( x  vH  A
)  i^i  B )  =  ( ( ( x  i^i  B )  vH  A )  i^i 
B ) )
2017oveq1d 5889 . . . . . . . 8  |-  ( x 
C_  B  ->  (
x  vH  ( A  i^i  B ) )  =  ( ( x  i^i 
B )  vH  ( A  i^i  B ) ) )
2119, 20eqeq12d 2310 . . . . . . 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 2631 . . . . 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 3212 . . . . . . 7  |-  ( x  =  y  ->  (
x  C_  B  <->  y  C_  B ) )
25 oveq1 5881 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  vH  A )  =  ( y  vH  A ) )
2625ineq1d 3382 . . . . . . . 8  |-  ( x  =  y  ->  (
( x  vH  A
)  i^i  B )  =  ( ( y  vH  A )  i^i 
B ) )
27 oveq1 5881 . . . . . . . 8  |-  ( x  =  y  ->  (
x  vH  ( A  i^i  B ) )  =  ( y  vH  ( A  i^i  B ) ) )
2826, 27eqeq12d 2310 . . . . . . 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 2777 . . . . 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 1632    e. wcel 1696   A.wral 2556    i^i cin 3164    C_ wss 3165   class class class wbr 4039  (class class class)co 5874   CHcch 21525    vH chj 21529    MH cmd 21562
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-i2m1 8821  ax-1ne0 8822  ax-rrecex 8825  ax-cnre 8826  ax-hilex 21595  ax-hfvadd 21596  ax-hv0cl 21599  ax-hfvmul 21601
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-map 6790  df-nn 9763  df-hlim 21568  df-sh 21802  df-ch 21817  df-md 22876
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