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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 3951 | . 2 ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶) |
3 | difeq12i.2 | . . 3 ⊢ 𝐶 = 𝐷 | |
4 | 3 | difeq2i 3952 | . 2 ⊢ (𝐵 ∖ 𝐶) = (𝐵 ∖ 𝐷) |
5 | 2, 4 | eqtri 2849 | 1 ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1658 ∖ cdif 3795 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rab 3126 df-dif 3801 |
This theorem is referenced by: difrab 4130 preddif 5945 uniioombllem4 23752 clwwlknclwwlkdif 27308 gtiso 30026 mthmpps 32025 zrdivrng 34294 isdrngo1 34297 pwfi2f1o 38509 salexct2 41348 dfnelbr2 42175 |
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