Theorem addneintr2d 8331

Description: Introducing a term on the right-hand side of a sum in a negated equality. Contrapositive of addcan2ad 8329. Consequence of addcan2d 8327. (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 )
addneintr2d.4	|- ( ph -> A =/= B )

Assertion

Ref	Expression
addneintr2d	|- ( ph -> ( A + C ) =/= ( B + C ) )

Proof of Theorem addneintr2d

Step	Hyp	Ref	Expression
1		addneintr2d.4	. 2 |- ( ph -> A =/= B )
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	addcan2d 8327	. . 3 |- ( ph -> ( ( A + C ) = ( B + C ) <-> A = B ) )
6	5	necon3bid 2441	. 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 2200   =/= wne 2400  (class class class)co 6000   CCcc 7993   + caddc 7998

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-resscn 8087  ax-1cn 8088  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-distr 8099  ax-i2m1 8100  ax-0id 8103  ax-rnegex 8104  ax-cnre 8106

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-iota 5277  df-fv 5325  df-ov 6003

This theorem is referenced by:  modsumfzodifsn  10613