Theorem subneintr2d 8310

Description: Introducing subtraction on both sides of a statement of inequality. Contrapositive of subcan2d 8306. (Contributed by David Moews, 28-Feb-2017.)

Hypotheses

Ref	Expression
negidd.1	|- ( ph -> A e. CC )
pncand.2	|- ( ph -> B e. CC )
subaddd.3	|- ( ph -> C e. CC )
subneintr2d.4	|- ( ph -> A =/= B )

Assertion

Ref	Expression
subneintr2d	|- ( ph -> ( A - C ) =/= ( B - C ) )

Proof of Theorem subneintr2d

Step	Hyp	Ref	Expression
1		subneintr2d.4	. 2 |- ( ph -> A =/= B )
2		negidd.1	. . . 4 |- ( ph -> A e. CC )
3		pncand.2	. . . 4 |- ( ph -> B e. CC )
4		subaddd.3	. . . 4 |- ( ph -> C e. CC )
5	2, 3, 4	subcan2ad 8309	. . 3 |- ( ph -> ( ( A - C ) = ( B - C ) <-> A = B ) )
6	5	necon3bid 2388	. 2 |- ( ph -> ( ( A - C ) =/= ( B - C ) <-> A =/= B ) )
7	1, 6	mpbird 167	1 |- ( ph -> ( A - C ) =/= ( B - C ) )

Syntax hints:   -> wi 4   e. wcel 2148   =/= wne 2347  (class class class)co 5872   CCcc 7806   - cmin 8124

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 4120  ax-pow 4173  ax-pr 4208  ax-setind 4535  ax-resscn 7900  ax-1cn 7901  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-addcom 7908  ax-addass 7910  ax-distr 7912  ax-i2m1 7913  ax-0id 7916  ax-rnegex 7917  ax-cnre 7919

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 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-opab 4064  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-iota 5177  df-fun 5217  df-fv 5223  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-sub 8126

This theorem is referenced by: (None)