Theorem pncan3oi 8388

Description: Subtraction and addition of equals. Almost but not exactly the same as pncan3i 8449 and pncan 8378, this order happens often when applying "operations to both sides" so create a theorem specifically for it. A deduction version of this is available as pncand 8484. (Contributed by David A. Wheeler, 11-Oct-2018.)

Hypotheses

Ref	Expression
pncan3oi.1	|- A e. CC
pncan3oi.2	|- B e. CC

Assertion

Ref	Expression
pncan3oi	|- ( ( A + B ) - B ) = A

Proof of Theorem pncan3oi

Step	Hyp	Ref	Expression
1		pncan3oi.1	. 2 |- A e. CC
2		pncan3oi.2	. 2 |- B e. CC
3		pncan 8378	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) - B ) = A )
4	1, 2, 3	mp2an 426	1 |- ( ( A + B ) - B ) = A

Syntax hints:   = wceq 1395   e. wcel 2200  (class class class)co 6013   CCcc 8023   + caddc 8028   - cmin 8343

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-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-setind 4633  ax-resscn 8117  ax-1cn 8118  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-addcom 8125  ax-addass 8127  ax-distr 8129  ax-i2m1 8130  ax-0id 8133  ax-rnegex 8134  ax-cnre 8136

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-iota 5284  df-fun 5326  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-sub 8345

This theorem is referenced by:  mvrraddi  8389  mvlladdi  8390  resqrexlemcalc1  11568  3dvds  12418