Theorem subadd 8523

Description: Relationship between subtraction and addition. (Contributed by NM, 20-Jan-1997.) (Revised by Mario Carneiro, 21-Dec-2013.)

Assertion

Ref	Expression
subadd	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴))

Proof of Theorem subadd

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		subval 8512	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) = (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴))
2	1	eqeq1d 2247	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) = 𝐶 ↔ (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) = 𝐶))
3	2	3adant3 1048	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐵) = 𝐶 ↔ (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) = 𝐶))
4		negeu 8511	. . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴)
5		oveq2 6087	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝐵 + 𝑥) = (𝐵 + 𝐶))
6	5	eqeq1d 2247	. . . . . 6 ⊢ (𝑥 = 𝐶 → ((𝐵 + 𝑥) = 𝐴 ↔ (𝐵 + 𝐶) = 𝐴))
7	6	riota2 6056	. . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ ∃!𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) → ((𝐵 + 𝐶) = 𝐴 ↔ (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) = 𝐶))
8	4, 7	sylan2 286	. . . 4 ⊢ ((𝐶 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ)) → ((𝐵 + 𝐶) = 𝐴 ↔ (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) = 𝐶))
9	8	3impb 1230	. . 3 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((𝐵 + 𝐶) = 𝐴 ↔ (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) = 𝐶))
10	9	3com13 1239	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐵 + 𝐶) = 𝐴 ↔ (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) = 𝐶))
11	3, 10	bitr4d 191	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ∃!wreu 2530  ℩crio 6031  (class class class)co 6079  ℂcc 8171   + caddc 8176   − cmin 8491

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 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 4247  ax-pow 4309  ax-pr 4344  ax-setind 4682  ax-resscn 8265  ax-1cn 8266  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-addass 8275  ax-distr 8277  ax-i2m1 8278  ax-0id 8281  ax-rnegex 8282  ax-cnre 8284

This theorem 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 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-iota 5335  df-fun 5377  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-sub 8493

This theorem is referenced by:  subadd2  8524  subsub23  8525  pncan  8526  pncan3  8528  addsubeq4  8535  subsub2  8548  renegcl  8581  subaddi  8607  subaddd  8649  fzen  10430  nn0ennn  10853  cos2t  12500  cos2tsin  12501  odd2np1  12623  divalgb  12675  sincosq1eq  15923  coskpi  15932