Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > negeqd | GIF version |
Description: Equality deduction for negatives. (Contributed by NM, 14-May-1999.) |
Ref | Expression |
---|---|
negeqd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
negeqd | ⊢ (𝜑 → -𝐴 = -𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negeqd.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | negeq 8099 | . 2 ⊢ (𝐴 = 𝐵 → -𝐴 = -𝐵) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → -𝐴 = -𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 -cneg 8078 |
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-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-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-rex 2454 df-v 2732 df-un 3125 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-iota 5158 df-fv 5204 df-ov 5853 df-neg 8080 |
This theorem is referenced by: negdi 8163 mulneg2 8302 mulm1 8306 eqord2 8390 mulreim 8510 apneg 8517 divnegap 8610 div2negap 8639 recgt0 8753 infrenegsupex 9540 supminfex 9543 mul2lt0rlt0 9703 ceilqval 10249 ceilid 10258 modqcyc2 10303 monoord2 10420 reneg 10819 imneg 10827 cjcj 10834 cjneg 10841 minmax 11180 minabs 11186 telfsumo2 11417 sinneg 11676 tannegap 11678 sincossq 11698 odd2np1 11819 oexpneg 11823 modgcd 11933 pcneg 12265 ivthdec 13337 limcimolemlt 13348 dvrecap 13392 sinperlem 13444 efimpi 13455 ptolemy 13460 lgsneg1 13641 ex-ceil 13682 |
Copyright terms: Public domain | W3C validator |