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Theorem acongeq12d 30510
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 6295 . . 3  |-  ( ph  ->  ( B  -  D
)  =  ( C  -  E ) )
43breq2d 4454 . 2  |-  ( ph  ->  ( A  ||  ( B  -  D )  <->  A 
||  ( C  -  E ) ) )
52negeqd 9805 . . . 4  |-  ( ph  -> 
-u D  =  -u E )
61, 5oveq12d 6295 . . 3  |-  ( ph  ->  ( B  -  -u D
)  =  ( C  -  -u E ) )
76breq2d 4454 . 2  |-  ( ph  ->  ( A  ||  ( B  -  -u D )  <-> 
A  ||  ( C  -  -u E ) ) )
84, 7orbi12d 709 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 368    = wceq 1374   class class class wbr 4442  (class class class)co 6277    - cmin 9796   -ucneg 9797    || cdivides 13838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-iota 5544  df-fv 5589  df-ov 6280  df-neg 9799
This theorem is referenced by:  acongrep  30511  jm2.26a  30537  jm2.26  30539
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