Theorem addcan2ad 8274

Description: Cancelling a term on the right-hand side of a sum in an equality. Consequence of addcan2d 8272. (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 )
addcan2ad.4	|- ( ph -> ( A + C ) = ( B + C ) )

Assertion

Ref	Expression
addcan2ad	|- ( ph -> A = B )

Proof of Theorem addcan2ad

Step	Hyp	Ref	Expression
1		addcan2ad.4	. 2 |- ( ph -> ( A + C ) = ( B + C ) )
2		addcand.1	. . 3 |- ( ph -> A e. CC )
3		addcand.2	. . 3 |- ( ph -> B e. CC )
4		addcand.3	. . 3 |- ( ph -> C e. CC )
5	2, 3, 4	addcan2d 8272	. 2 |- ( ph -> ( ( A + C ) = ( B + C ) <-> A = B ) )
6	1, 5	mpbid 147	1 |- ( ph -> A = B )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2177  (class class class)co 5956   CCcc 7938   + caddc 7943

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-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188  ax-resscn 8032  ax-1cn 8033  ax-icn 8035  ax-addcl 8036  ax-addrcl 8037  ax-mulcl 8038  ax-addcom 8040  ax-addass 8042  ax-distr 8044  ax-i2m1 8045  ax-0id 8048  ax-rnegex 8049  ax-cnre 8051

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-br 4051  df-iota 5240  df-fv 5287  df-ov 5959

This theorem is referenced by: (None)