Theorem addneintrd 7199

Description: Introducing a term on the left-hand side of a sum in a negated equality. Contrapositive of addcanad 7197. Consequence of addcand 7195. (Contributed by David Moews, 28-Feb-2017.)

Hypotheses

Ref	Expression
addcand.1	|- ( ph -> A e. CC )
addcand.2	|- ( ph -> B e. CC )
addcand.3	|- ( ph -> C e. CC )
addneintrd.4	|- ( ph -> B =/= C )

Assertion

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

Proof of Theorem addneintrd

Step	Hyp	Ref	Expression
1		addneintrd.4	. 2 |- ( ph -> B =/= C )
2		addcand.1	. . . 4 |- ( ph -> A e. CC )
3		addcand.2	. . . 4 |- ( ph -> B e. CC )
4		addcand.3	. . . 4 |- ( ph -> C e. CC )
5	2, 3, 4	addcand 7195	. . 3 |- ( ph -> ( ( A + B ) = ( A + C ) <-> B = C ) )
6	5	necon3bid 2246	. 2 |- ( ph -> ( ( A + B ) =/= ( A + C ) <-> B =/= C ) )
7	1, 6	mpbird 156	1 |- ( ph -> ( A + B ) =/= ( A + C ) )

Syntax hints:   -> wi 4   e. wcel 1393   =/= wne 2204  (class class class)co 5512   CCcc 6887   + caddc 6892

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-resscn 6976  ax-1cn 6977  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515

This theorem is referenced by: (None)