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Mirrors > Home > ILE Home > Th. List > difeq2d | GIF version |
Description: Deduction adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.) |
Ref | Expression |
---|---|
difeq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
difeq2d | ⊢ (𝜑 → (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difeq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | difeq2 3239 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ∖ cdif 3118 |
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-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-11 1499 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-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-ral 2453 df-rab 2457 df-dif 3123 |
This theorem is referenced by: difeq12d 3246 phplem3 6832 phplem4 6833 phplem3g 6834 phplem4dom 6840 phplem4on 6845 fidifsnen 6848 xpfi 6907 sbthlem2 6935 sbthlemi3 6936 isbth 6944 ismkvnex 7131 setsvalg 12446 setsvala 12447 exmid1stab 14033 |
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