Theorem addneintr2d 8148

Description: Introducing a term on the right-hand side of a sum in a negated equality. Contrapositive of addcan2ad 8146. Consequence of addcan2d 8144. (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 8144	. . 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 5877   CCcc 7811   + caddc 7816

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-ext 2159  ax-resscn 7905  ax-1cn 7906  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-iota 5180  df-fv 5226  df-ov 5880

This theorem is referenced by:  modsumfzodifsn  10398