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Theorem acongeq12d 27066
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 5876 . . 3  |-  ( ph  ->  ( B  -  D
)  =  ( C  -  E ) )
43breq2d 4035 . 2  |-  ( ph  ->  ( A  ||  ( B  -  D )  <->  A 
||  ( C  -  E ) ) )
52negeqd 9046 . . . 4  |-  ( ph  -> 
-u D  =  -u E )
61, 5oveq12d 5876 . . 3  |-  ( ph  ->  ( B  -  -u D
)  =  ( C  -  -u E ) )
76breq2d 4035 . 2  |-  ( ph  ->  ( A  ||  ( B  -  -u D )  <-> 
A  ||  ( C  -  -u E ) ) )
84, 7orbi12d 690 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 176    \/ wo 357    = wceq 1623   class class class wbr 4023  (class class class)co 5858    - cmin 9037   -ucneg 9038    || cdivides 12531
This theorem is referenced by:  acongrep  27067  jm2.26a  27093  jm2.26  27095
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-neg 9040
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