Theorem addneintrd 8162

Description: Introducing a term on the left-hand side of a sum in a negated equality. Contrapositive of addcanad 8160. Consequence of addcand 8158. (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 8158	. . 3 |- ( ph -> ( ( A + B ) = ( A + C ) <-> B = C ) )
6	5	necon3bid 2400	. 2 |- ( ph -> ( ( A + B ) =/= ( A + C ) <-> B =/= C ) )
7	1, 6	mpbird 167	1 |- ( ph -> ( A + B ) =/= ( A + C ) )

Syntax hints:   -> wi 4   e. wcel 2159   =/= wne 2359  (class class class)co 5890   CCcc 7826   + caddc 7831

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170  ax-resscn 7920  ax-1cn 7921  ax-icn 7923  ax-addcl 7924  ax-addrcl 7925  ax-mulcl 7926  ax-addcom 7928  ax-addass 7930  ax-distr 7932  ax-i2m1 7933  ax-0id 7936  ax-rnegex 7937  ax-cnre 7939

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-ral 2472  df-rex 2473  df-v 2753  df-un 3147  df-in 3149  df-ss 3156  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-br 4018  df-iota 5192  df-fv 5238  df-ov 5893

This theorem is referenced by: (None)