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Theorem acongeq12d 35278
Description: Substitution deduction for alternating congruence. (Contributed by Stefan O'Rear, 3-Oct-2014.)
Hypotheses
Ref Expression
acongeq12d.1  |-  ( ph  ->  B  =  C )
acongeq12d.2  |-  ( ph  ->  D  =  E )
Assertion
Ref Expression
acongeq12d  |-  ( ph  ->  ( ( A  ||  ( B  -  D
)  \/  A  ||  ( B  -  -u D
) )  <->  ( A  ||  ( C  -  E
)  \/  A  ||  ( C  -  -u E
) ) ) )

Proof of Theorem acongeq12d
StepHypRef Expression
1 acongeq12d.1 . . . 4  |-  ( ph  ->  B  =  C )
2 acongeq12d.2 . . . 4  |-  ( ph  ->  D  =  E )
31, 2oveq12d 6296 . . 3  |-  ( ph  ->  ( B  -  D
)  =  ( C  -  E ) )
43breq2d 4407 . 2  |-  ( ph  ->  ( A  ||  ( B  -  D )  <->  A 
||  ( C  -  E ) ) )
52negeqd 9850 . . . 4  |-  ( ph  -> 
-u D  =  -u E )
61, 5oveq12d 6296 . . 3  |-  ( ph  ->  ( B  -  -u D
)  =  ( C  -  -u E ) )
76breq2d 4407 . 2  |-  ( ph  ->  ( A  ||  ( B  -  -u D )  <-> 
A  ||  ( C  -  -u E ) ) )
84, 7orbi12d 708 1  |-  ( ph  ->  ( ( A  ||  ( B  -  D
)  \/  A  ||  ( B  -  -u D
) )  <->  ( A  ||  ( C  -  E
)  \/  A  ||  ( C  -  -u E
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 366    = wceq 1405   class class class wbr 4395  (class class class)co 6278    - cmin 9841   -ucneg 9842    || cdvds 14195
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-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-iota 5533  df-fv 5577  df-ov 6281  df-neg 9844
This theorem is referenced by:  acongrep  35279  jm2.26a  35304  jm2.26  35306
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