Metamath Proof Explorer

Theorem equncomiVD

Description: Inference form of equncom . The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. equncomi is equncomiVD without virtual deductions and was automatically derived from equncomiVD .

h1::	\|- A = ( B u. C )
qed:1:	\|- A = ( C u. B )

(Contributed by Alan Sare, 18-Feb-2012) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	equncomiVD.1	⊢ 𝐴 = ( 𝐵 ∪ 𝐶 )
	Assertion	equncomiVD	⊢ 𝐴 = ( 𝐶 ∪ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		equncomiVD.1	⊢ 𝐴 = ( 𝐵 ∪ 𝐶 )
2		equncom	⊢ ( 𝐴 = ( 𝐵 ∪ 𝐶 ) ↔ 𝐴 = ( 𝐶 ∪ 𝐵 ) )
3	2	biimpi	⊢ ( 𝐴 = ( 𝐵 ∪ 𝐶 ) → 𝐴 = ( 𝐶 ∪ 𝐵 ) )
4	1 3	e0a	⊢ 𝐴 = ( 𝐶 ∪ 𝐵 )