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Metamath Proof Explorer

Theorem equncom

Description: If a class equals the union of two other classes, then it equals the union of those two classes commuted. equncom was automatically derived from equncomVD using the tools program translate__without__overwriting.cmd and minimizing. (Contributed by Alan Sare, 18-Feb-2012)

		Ref	Expression
	Assertion	equncom	⊢ ( 𝐴 = ( 𝐵 ∪ 𝐶 ) ↔ 𝐴 = ( 𝐶 ∪ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		uncom	⊢ ( 𝐵 ∪ 𝐶 ) = ( 𝐶 ∪ 𝐵 )
2	1	eqeq2i	⊢ ( 𝐴 = ( 𝐵 ∪ 𝐶 ) ↔ 𝐴 = ( 𝐶 ∪ 𝐵 ) )