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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 4155 was automatically derived from equncomVD 43629 using the tools program translate_without_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 4154 | . 2 ⊢ (𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵) | |
2 | 1 | eqeq2i 2746 | 1 ⊢ (𝐴 = (𝐵 ∪ 𝐶) ↔ 𝐴 = (𝐶 ∪ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1542 ∪ cun 3947 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-un 3954 |
This theorem is referenced by: equncomi 4156 equncomiVD 43630 |
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