Theorem negned 8242

Description: If two complex numbers are unequal, so are their negatives. Contrapositive of neg11d 8257. (Contributed by David Moews, 28-Feb-2017.)

Hypotheses

Ref	Expression
negidd.1	|- ( ph -> A e. CC )
negned.2	|- ( ph -> B e. CC )
negned.3	|- ( ph -> A =/= B )

Assertion

Ref	Expression
negned	|- ( ph -> -u A =/= -u B )

Proof of Theorem negned

Step	Hyp	Ref	Expression
1		negned.3	. 2 |- ( ph -> A =/= B )
2		negidd.1	. . . 4 |- ( ph -> A e. CC )
3		negned.2	. . . 4 |- ( ph -> B e. CC )
4	2, 3	neg11ad 8241	. . 3 |- ( ph -> ( -u A = -u B <-> A = B ) )
5	4	necon3bid 2388	. 2 |- ( ph -> ( -u A =/= -u B <-> A =/= B ) )
6	1, 5	mpbird 167	1 |- ( ph -> -u A =/= -u B )

Syntax hints:   -> wi 4   e. wcel 2148   =/= wne 2347   CCcc 7787   -ucneg 8106

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 4118  ax-pow 4171  ax-pr 4205  ax-setind 4532  ax-resscn 7881  ax-1cn 7882  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-addcom 7889  ax-addass 7891  ax-distr 7893  ax-i2m1 7894  ax-0id 7897  ax-rnegex 7898  ax-cnre 7900

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 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-iota 5173  df-fun 5213  df-fv 5219  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-sub 8107  df-neg 8108

This theorem is referenced by: (None)