Theorem subaddeqd 8223

Description: Transfer two terms of a subtraction to an addition in an equality. (Contributed by Thierry Arnoux, 2-Feb-2020.)

Hypotheses

Ref	Expression
subaddeqd.a	|- ( ph -> A e. CC )
subaddeqd.b	|- ( ph -> B e. CC )
subaddeqd.c	|- ( ph -> C e. CC )
subaddeqd.d	|- ( ph -> D e. CC )
subaddeqd.1	|- ( ph -> ( A + B ) = ( C + D ) )

Assertion

Ref	Expression
subaddeqd	|- ( ph -> ( A - D ) = ( C - B ) )

Proof of Theorem subaddeqd

Step	Hyp	Ref	Expression
1		subaddeqd.1	. . . 4 |- ( ph -> ( A + B ) = ( C + D ) )
2	1	oveq1d 5829	. . 3 |- ( ph -> ( ( A + B ) - ( D + B ) ) = ( ( C + D ) - ( D + B ) ) )
3		subaddeqd.c	. . . . 5 |- ( ph -> C e. CC )
4		subaddeqd.d	. . . . 5 |- ( ph -> D e. CC )
5	3, 4	addcomd 8005	. . . 4 |- ( ph -> ( C + D ) = ( D + C ) )
6	5	oveq1d 5829	. . 3 |- ( ph -> ( ( C + D ) - ( D + B ) ) = ( ( D + C ) - ( D + B ) ) )
7	2, 6	eqtrd 2187	. 2 |- ( ph -> ( ( A + B ) - ( D + B ) ) = ( ( D + C ) - ( D + B ) ) )
8		subaddeqd.a	. . 3 |- ( ph -> A e. CC )
9		subaddeqd.b	. . 3 |- ( ph -> B e. CC )
10	8, 4, 9	pnpcan2d 8203	. 2 |- ( ph -> ( ( A + B ) - ( D + B ) ) = ( A - D ) )
11	4, 3, 9	pnpcand 8202	. 2 |- ( ph -> ( ( D + C ) - ( D + B ) ) = ( C - B ) )
12	7, 10, 11	3eqtr3d 2195	1 |- ( ph -> ( A - D ) = ( C - B ) )

Syntax hints:   -> wi 4   = wceq 1332   e. wcel 2125  (class class class)co 5814   CCcc 7709   + caddc 7714   - cmin 8025

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164  ax-setind 4490  ax-resscn 7803  ax-1cn 7804  ax-icn 7806  ax-addcl 7807  ax-addrcl 7808  ax-mulcl 7809  ax-addcom 7811  ax-addass 7813  ax-distr 7815  ax-i2m1 7816  ax-0id 7819  ax-rnegex 7820  ax-cnre 7822

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-ral 2437  df-rex 2438  df-reu 2439  df-rab 2441  df-v 2711  df-sbc 2934  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-br 3962  df-opab 4022  df-id 4248  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-iota 5128  df-fun 5165  df-fv 5171  df-riota 5770  df-ov 5817  df-oprab 5818  df-mpo 5819  df-sub 8027

This theorem is referenced by: (None)