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Theorem acongeq12d 27046
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 6101 . . 3  |-  ( ph  ->  ( B  -  D
)  =  ( C  -  E ) )
43breq2d 4226 . 2  |-  ( ph  ->  ( A  ||  ( B  -  D )  <->  A 
||  ( C  -  E ) ) )
52negeqd 9302 . . . 4  |-  ( ph  -> 
-u D  =  -u E )
61, 5oveq12d 6101 . . 3  |-  ( ph  ->  ( B  -  -u D
)  =  ( C  -  -u E ) )
76breq2d 4226 . 2  |-  ( ph  ->  ( A  ||  ( B  -  -u D )  <-> 
A  ||  ( C  -  -u E ) ) )
84, 7orbi12d 692 1  |-  ( ph  ->  ( ( A  ||  ( B  -  D
)  \/  A  ||  ( B  -  -u D
) )  <->  ( A  ||  ( C  -  E
)  \/  A  ||  ( C  -  -u E
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    \/ wo 359    = wceq 1653   class class class wbr 4214  (class class class)co 6083    - cmin 9293   -ucneg 9294    || cdivides 12854
This theorem is referenced by:  acongrep  27047  jm2.26a  27073  jm2.26  27075
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-ov 6086  df-neg 9296
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