Theorem acongeq12d 37546

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

Step	Hyp	Ref	Expression
1		acongeq12d.1	. . . 4 |- ( ph -> B = C )
2		acongeq12d.2	. . . 4 |- ( ph -> D = E )
3	1, 2	oveq12d 6668	. . 3 |- ( ph -> ( B - D ) = ( C - E ) )
4	3	breq2d 4665	. 2 |- ( ph -> ( A || ( B - D ) <-> A || ( C - E ) ) )
5	2	negeqd 10275	. . . 4 |- ( ph -> -u D = -u E )
6	1, 5	oveq12d 6668	. . 3 |- ( ph -> ( B - -u D ) = ( C - -u E ) )
7	6	breq2d 4665	. 2 |- ( ph -> ( A || ( B - -u D ) <-> A || ( C - -u E ) ) )
8	4, 7	orbi12d 746	1 |- ( ph -> ( ( A || ( B - D ) \/ A || ( B - -u D ) ) <-> ( A || ( C - E ) \/ A || ( C - -u E ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   = wceq 1483   class class class wbr 4653  (class class class)co 6650   - cmin 10266   -ucneg 10267   || cdvds 14983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-neg 10269

This theorem is referenced by:  acongrep  37547  jm2.26a  37567  jm2.26  37569