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Mirrors > Home > ILE Home > Th. List > difeq1d | GIF version |
Description: Deduction adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.) |
Ref | Expression |
---|---|
difeq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
difeq1d | ⊢ (𝜑 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difeq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | difeq1 3246 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∖ cdif 3126 |
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-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rab 2464 df-dif 3131 |
This theorem is referenced by: difeq12d 3254 diftpsn3 3733 phplem4 6854 phplem3g 6855 phplem4on 6866 en2other2 7194 isstruct2im 12466 isstruct2r 12467 setsfun0 12492 ptex 12707 cldval 13530 difopn 13539 cnclima 13654 |
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