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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 4121 | . 2 ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶) | 
| 3 | difeq12i.2 | . . 3 ⊢ 𝐶 = 𝐷 | |
| 4 | 3 | difeq2i 4122 | . 2 ⊢ (𝐵 ∖ 𝐶) = (𝐵 ∖ 𝐷) | 
| 5 | 2, 4 | eqtri 2764 | 1 ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ∖ cdif 3947 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-dif 3953 | 
| This theorem is referenced by: indifdir 4294 difrab 4317 resdifdi 6255 resdifdir 6256 preddif 6349 infdju1 10231 uniioombllem4 25622 new0 27914 clwwlknclwwlkdif 29999 gtiso 32711 satffunlem2lem2 35412 mthmpps 35588 zrdivrng 37961 isdrngo1 37964 pwfi2f1o 43113 salexct2 46359 dfnelbr2 47290 | 
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