Theorem pncan3oi 8549

Description: Subtraction and addition of equals. Almost but not exactly the same as pncan3i 8610 and pncan 8539, 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 8645. (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 8539	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) - B ) = A )
4	1, 2, 3	mp2an 430	1 |- ( ( A + B ) - B ) = A

This proof depends on syntax axioms:   = wceq 1402   e. wcel 2209  (class class class)co 6088   CCcc 8183   + caddc 8188   - cmin 8504

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-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-setind 4686  ax-resscn 8277  ax-1cn 8278  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-distr 8289  ax-i2m1 8290  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-iota 5340  df-fun 5382  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-sub 8506

This theorem is used by:  mvrraddi  8550  mvlladdi  8551  resqrexlemcalc1  11802  3dvds  12656  bposlem6  16316  bposlem8  16318