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Mirrors > Home > MPE Home > Th. List > difeq12i | Structured version Visualization version GIF version |
Description: Equality inference for class difference. (Contributed by NM, 29-Aug-2004.) |
Ref | Expression |
---|---|
difeq1i.1 | ⊢ 𝐴 = 𝐵 |
difeq12i.2 | ⊢ 𝐶 = 𝐷 |
Ref | Expression |
---|---|
difeq12i | ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difeq1i.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
2 | 1 | difeq1i 3686 | . 2 ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶) |
3 | difeq12i.2 | . . 3 ⊢ 𝐶 = 𝐷 | |
4 | 3 | difeq2i 3687 | . 2 ⊢ (𝐵 ∖ 𝐶) = (𝐵 ∖ 𝐷) |
5 | 2, 4 | eqtri 2632 | 1 ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∖ cdif 3537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rab 2905 df-dif 3543 |
This theorem is referenced by: difrab 3860 preddif 5622 uniioombllem4 23160 gtiso 28861 mthmpps 30733 zrdivrng 32922 isdrngo1 32925 pwfi2f1o 36684 salexct2 39233 |
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