Theorem addneintrd 7670

Description: Introducing a term on the left-hand side of a sum in a negated equality. Contrapositive of addcanad 7668. Consequence of addcand 7666. (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 7666	. . 3 |- ( ph -> ( ( A + B ) = ( A + C ) <-> B = C ) )
6	5	necon3bid 2296	. 2 |- ( ph -> ( ( A + B ) =/= ( A + C ) <-> B =/= C ) )
7	1, 6	mpbird 165	1 |- ( ph -> ( A + B ) =/= ( A + C ) )

Syntax hints:   -> wi 4   e. wcel 1438   =/= wne 2255  (class class class)co 5652   CCcc 7348   + caddc 7353

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-resscn 7437  ax-1cn 7438  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-addcom 7445  ax-addass 7447  ax-distr 7449  ax-i2m1 7450  ax-0id 7453  ax-rnegex 7454  ax-cnre 7456

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-iota 4980  df-fv 5023  df-ov 5655

This theorem is referenced by: (None)