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Mirrors > Home > MPE Home > Th. List > symdifeq1 | Structured version Visualization version GIF version |
Description: Equality theorem for symmetric difference. (Contributed by Scott Fenton, 24-Apr-2012.) |
Ref | Expression |
---|---|
symdifeq1 | ⊢ (𝐴 = 𝐵 → (𝐴 △ 𝐶) = (𝐵 △ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difeq1 4091 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶)) | |
2 | difeq2 4092 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵)) | |
3 | 1, 2 | uneq12d 4139 | . 2 ⊢ (𝐴 = 𝐵 → ((𝐴 ∖ 𝐶) ∪ (𝐶 ∖ 𝐴)) = ((𝐵 ∖ 𝐶) ∪ (𝐶 ∖ 𝐵))) |
4 | df-symdif 4218 | . 2 ⊢ (𝐴 △ 𝐶) = ((𝐴 ∖ 𝐶) ∪ (𝐶 ∖ 𝐴)) | |
5 | df-symdif 4218 | . 2 ⊢ (𝐵 △ 𝐶) = ((𝐵 ∖ 𝐶) ∪ (𝐶 ∖ 𝐵)) | |
6 | 3, 4, 5 | 3eqtr4g 2881 | 1 ⊢ (𝐴 = 𝐵 → (𝐴 △ 𝐶) = (𝐵 △ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∖ cdif 3932 ∪ cun 3933 △ csymdif 4217 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-symdif 4218 |
This theorem is referenced by: symdifeq2 4221 |
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