Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > mvlladdd | GIF version |
Description: Move LHS left addition to RHS. (Contributed by David A. Wheeler, 15-Oct-2018.) |
Ref | Expression |
---|---|
mvlraddd.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
mvlraddd.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
mvlraddd.3 | ⊢ (𝜑 → (𝐴 + 𝐵) = 𝐶) |
Ref | Expression |
---|---|
mvlladdd | ⊢ (𝜑 → 𝐵 = (𝐶 − 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mvlraddd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
2 | mvlraddd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
3 | 1, 2 | pncand 8077 | . 2 ⊢ (𝜑 → ((𝐵 + 𝐴) − 𝐴) = 𝐵) |
4 | 2, 1 | addcomd 7916 | . . . 4 ⊢ (𝜑 → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
5 | mvlraddd.3 | . . . 4 ⊢ (𝜑 → (𝐴 + 𝐵) = 𝐶) | |
6 | 4, 5 | eqtr3d 2174 | . . 3 ⊢ (𝜑 → (𝐵 + 𝐴) = 𝐶) |
7 | 6 | oveq1d 5789 | . 2 ⊢ (𝜑 → ((𝐵 + 𝐴) − 𝐴) = (𝐶 − 𝐴)) |
8 | 3, 7 | eqtr3d 2174 | 1 ⊢ (𝜑 → 𝐵 = (𝐶 − 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 (class class class)co 5774 ℂcc 7621 + caddc 7626 − cmin 7936 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-setind 4452 ax-resscn 7715 ax-1cn 7716 ax-icn 7718 ax-addcl 7719 ax-addrcl 7720 ax-mulcl 7721 ax-addcom 7723 ax-addass 7725 ax-distr 7727 ax-i2m1 7728 ax-0id 7731 ax-rnegex 7732 ax-cnre 7734 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-sub 7938 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |