Theorem subeqrev 8333

Description: Reverse the order of subtraction in an equality. (Contributed by Scott Fenton, 8-Jul-2013.)

Assertion

Ref	Expression
subeqrev	|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A - B ) = ( C - D ) <-> ( B - A ) = ( D - C ) ) )

Proof of Theorem subeqrev

Step	Hyp	Ref	Expression
1		subcl 8156	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
2		subcl 8156	. . 3 |- ( ( C e. CC /\ D e. CC ) -> ( C - D ) e. CC )
3		neg11 8208	. . 3 |- ( ( ( A - B ) e. CC /\ ( C - D ) e. CC ) -> ( -u ( A - B ) = -u ( C - D ) <-> ( A - B ) = ( C - D ) ) )
4	1, 2, 3	syl2an 289	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( -u ( A - B ) = -u ( C - D ) <-> ( A - B ) = ( C - D ) ) )
5		negsubdi2 8216	. . 3 |- ( ( A e. CC /\ B e. CC ) -> -u ( A - B ) = ( B - A ) )
6		negsubdi2 8216	. . 3 |- ( ( C e. CC /\ D e. CC ) -> -u ( C - D ) = ( D - C ) )
7	5, 6	eqeqan12d 2193	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( -u ( A - B ) = -u ( C - D ) <-> ( B - A ) = ( D - C ) ) )
8	4, 7	bitr3d 190	1 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A - B ) = ( C - D ) <-> ( B - A ) = ( D - C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148  (class class class)co 5875   CCcc 7809   - cmin 8128   -ucneg 8129

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-setind 4537  ax-resscn 7903  ax-1cn 7904  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addcom 7911  ax-addass 7913  ax-distr 7915  ax-i2m1 7916  ax-0id 7919  ax-rnegex 7920  ax-cnre 7922

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-iota 5179  df-fun 5219  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-sub 8130  df-neg 8131

This theorem is referenced by: (None)