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Theorem assraddsubd 8276
Description: Associate RHS addition-subtraction. (Contributed by David A. Wheeler, 15-Oct-2018.)
Hypotheses
Ref Expression
assraddsubd.1  |-  ( ph  ->  B  e.  CC )
assraddsubd.2  |-  ( ph  ->  C  e.  CC )
assraddsubd.3  |-  ( ph  ->  D  e.  CC )
assraddsubd.4  |-  ( ph  ->  A  =  ( ( B  +  C )  -  D ) )
Assertion
Ref Expression
assraddsubd  |-  ( ph  ->  A  =  ( B  +  ( C  -  D ) ) )

Proof of Theorem assraddsubd
StepHypRef Expression
1 assraddsubd.4 . 2  |-  ( ph  ->  A  =  ( ( B  +  C )  -  D ) )
2 assraddsubd.1 . . 3  |-  ( ph  ->  B  e.  CC )
3 assraddsubd.2 . . 3  |-  ( ph  ->  C  e.  CC )
4 assraddsubd.3 . . 3  |-  ( ph  ->  D  e.  CC )
52, 3, 4addsubassd 8239 . 2  |-  ( ph  ->  ( ( B  +  C )  -  D
)  =  ( B  +  ( C  -  D ) ) )
61, 5eqtrd 2203 1  |-  ( ph  ->  A  =  ( B  +  ( C  -  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1348    e. wcel 2141  (class class class)co 5851   CCcc 7761    + caddc 7766    - cmin 8079
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-setind 4519  ax-resscn 7855  ax-1cn 7856  ax-icn 7858  ax-addcl 7859  ax-addrcl 7860  ax-mulcl 7861  ax-addcom 7863  ax-addass 7865  ax-distr 7867  ax-i2m1 7868  ax-0id 7871  ax-rnegex 7872  ax-cnre 7874
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fv 5204  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-sub 8081
This theorem is referenced by:  tangtx  13514
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