Theorem addneintr2d 8108

Description: Introducing a term on the right-hand side of a sum in a negated equality. Contrapositive of addcan2ad 8106. Consequence of addcan2d 8104. (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 8104	. . 3 |- ( ph -> ( ( A + C ) = ( B + C ) <-> A = B ) )
6	5	necon3bid 2381	. 2 |- ( ph -> ( ( A + C ) =/= ( B + C ) <-> A =/= B ) )
7	1, 6	mpbird 166	1 |- ( ph -> ( A + C ) =/= ( B + C ) )

Syntax hints:   -> wi 4   e. wcel 2141   =/= wne 2340  (class class class)co 5853   CCcc 7772   + caddc 7777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-resscn 7866  ax-1cn 7867  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-fv 5206  df-ov 5856

This theorem is referenced by:  modsumfzodifsn  10352