Theorem equncomVD 45835

Description: If a class equals the union of two other classes, then it equals the union of those two classes commuted. 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. equncom 4106 is equncomVD 45835 without virtual deductions and was automatically derived from equncomVD 45835.

1::	⊢ (   𝐴 = (𝐵 ∪ 𝐶)   ▶   𝐴 = (𝐵 ∪ 𝐶)   )
2::	⊢ (𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵)
3:1,2:	⊢ (   𝐴 = (𝐵 ∪ 𝐶)   ▶   𝐴 = (𝐶 ∪ 𝐵)   )
4:3:	⊢ (𝐴 = (𝐵 ∪ 𝐶) → 𝐴 = (𝐶 ∪ 𝐵))
5::	⊢ (   𝐴 = (𝐶 ∪ 𝐵)   ▶   𝐴 = (𝐶 ∪ 𝐵)   )
6:5,2:	⊢ (   𝐴 = (𝐶 ∪ 𝐵)   ▶   𝐴 = (𝐵 ∪ 𝐶)   )
7:6:	⊢ (𝐴 = (𝐶 ∪ 𝐵) → 𝐴 = (𝐵 ∪ 𝐶))
8:4,7:	⊢ (𝐴 = (𝐵 ∪ 𝐶) ↔ 𝐴 = (𝐶 ∪ 𝐵))

(Contributed by Alan Sare, 17-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
equncomVD	⊢ (𝐴 = (𝐵 ∪ 𝐶) ↔ 𝐴 = (𝐶 ∪ 𝐵))

Proof of Theorem equncomVD

Step	Hyp	Ref	Expression
1		idn1 45542	. . . 4 ⊢ (   𝐴 = (𝐵 ∪ 𝐶)   ▶   𝐴 = (𝐵 ∪ 𝐶)   )
2		uncom 4105	. . . 4 ⊢ (𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵)
3		eqeq1 2765	. . . . 5 ⊢ (𝐴 = (𝐵 ∪ 𝐶) → (𝐴 = (𝐶 ∪ 𝐵) ↔ (𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵)))
4	3	biimprd 251	. . . 4 ⊢ (𝐴 = (𝐵 ∪ 𝐶) → ((𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵) → 𝐴 = (𝐶 ∪ 𝐵)))
5	1, 2, 4	e10 45662	. . 3 ⊢ (   𝐴 = (𝐵 ∪ 𝐶)   ▶   𝐴 = (𝐶 ∪ 𝐵)   )
6	5	in1 45539	. 2 ⊢ (𝐴 = (𝐵 ∪ 𝐶) → 𝐴 = (𝐶 ∪ 𝐵))
7		idn1 45542	. . . 4 ⊢ (   𝐴 = (𝐶 ∪ 𝐵)   ▶   𝐴 = (𝐶 ∪ 𝐵)   )
8		eqeq2 2773	. . . . 5 ⊢ ((𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵) → (𝐴 = (𝐵 ∪ 𝐶) ↔ 𝐴 = (𝐶 ∪ 𝐵)))
9	8	biimprcd 253	. . . 4 ⊢ (𝐴 = (𝐶 ∪ 𝐵) → ((𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵) → 𝐴 = (𝐵 ∪ 𝐶)))
10	7, 2, 9	e10 45662	. . 3 ⊢ (   𝐴 = (𝐶 ∪ 𝐵)   ▶   𝐴 = (𝐵 ∪ 𝐶)   )
11	10	in1 45539	. 2 ⊢ (𝐴 = (𝐶 ∪ 𝐵) → 𝐴 = (𝐵 ∪ 𝐶))
12	6, 11	impbii 212	1 ⊢ (𝐴 = (𝐵 ∪ 𝐶) ↔ 𝐴 = (𝐶 ∪ 𝐵))

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ∪ cun 3897

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-un 3904  df-vd1 45538

This theorem is used by: (None)