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Mirrors > Home > MPE Home > Th. List > equncom | Structured version Visualization version GIF version |
Description: If a class equals the union of two other classes, then it equals the union of those two classes commuted. equncom 3720 was automatically derived from equncomVD 38126 using the tools program translatewithout_overwriting.cmd and minimizing. (Contributed by Alan Sare, 18-Feb-2012.) |
Ref | Expression |
---|---|
equncom | ⊢ (𝐴 = (𝐵 ∪ 𝐶) ↔ 𝐴 = (𝐶 ∪ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uncom 3719 | . 2 ⊢ (𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵) | |
2 | 1 | eqeq2i 2622 | 1 ⊢ (𝐴 = (𝐵 ∪ 𝐶) ↔ 𝐴 = (𝐶 ∪ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∪ cun 3538 |
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-v 3175 df-un 3545 |
This theorem is referenced by: equncomi 3721 equncomiVD 38127 |
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