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Mirrors > Home > NFE Home > Th. List > equncom | GIF version |
Description: If a class equals the union of two other classes, then it equals the union of those two classes commuted. equncom 3409 was automatically derived from equncomVD in set.mm 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 3408 | . 2 ⊢ (B ∪ C) = (C ∪ B) | |
2 | 1 | eqeq2i 2363 | 1 ⊢ (A = (B ∪ C) ↔ A = (C ∪ B)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 = wceq 1642 ∪ cun 3207 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-un 3214 |
This theorem is referenced by: equncomi 3410 |
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