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Mirrors > Home > MPE Home > Th. List > difeq12 | Structured version Visualization version GIF version |
Description: Equality theorem for class difference. (Contributed by FL, 31-Aug-2009.) |
Ref | Expression |
---|---|
difeq12 | ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difeq1 4110 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶)) | |
2 | difeq2 4111 | . 2 ⊢ (𝐶 = 𝐷 → (𝐵 ∖ 𝐶) = (𝐵 ∖ 𝐷)) | |
3 | 1, 2 | sylan9eq 2786 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∖ cdif 3940 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-rab 3427 df-dif 3946 |
This theorem is referenced by: resdif 6847 |
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