Theorem addneintrd 8514

Description: Introducing a term on the left-hand side of a sum in a negated equality. Contrapositive of addcanad 8512. Consequence of addcand 8510. (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 8510	. . 3 |- ( ph -> ( ( A + B ) = ( A + C ) <-> B = C ) )
6	5	necon3bid 2461	. 2 |- ( ph -> ( ( A + B ) =/= ( A + C ) <-> B =/= C ) )
7	1, 6	mpbird 167	1 |- ( ph -> ( A + B ) =/= ( A + C ) )

This proof depends on syntax axioms:   -> wi 4   e. wcel 2209   =/= wne 2420  (class class class)co 6085   CCcc 8177   + caddc 8182

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-resscn 8271  ax-1cn 8272  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-iota 5337  df-fv 5385  df-ov 6088

This theorem is used by: (None)