Theorem equncomVD 41574

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 4081 is equncomVD 41574 without virtual deductions and was automatically derived from equncomVD 41574.

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 41280	. . . 4 ⊢ (   𝐴 = (𝐵 ∪ 𝐶)   ▶   𝐴 = (𝐵 ∪ 𝐶)   )
2		uncom 4080	. . . 4 ⊢ (𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵)
3		eqeq1 2802	. . . . 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 41400	. . 3 ⊢ (   𝐴 = (𝐵 ∪ 𝐶)   ▶   𝐴 = (𝐶 ∪ 𝐵)   )
6	5	in1 41277	. 2 ⊢ (𝐴 = (𝐵 ∪ 𝐶) → 𝐴 = (𝐶 ∪ 𝐵))
7		idn1 41280	. . . 4 ⊢ (   𝐴 = (𝐶 ∪ 𝐵)   ▶   𝐴 = (𝐶 ∪ 𝐵)   )
8		eqeq2 2810	. . . . 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 41400	. . 3 ⊢ (   𝐴 = (𝐶 ∪ 𝐵)   ▶   𝐴 = (𝐵 ∪ 𝐶)   )
11	10	in1 41277	. 2 ⊢ (𝐴 = (𝐶 ∪ 𝐵) → 𝐴 = (𝐵 ∪ 𝐶))
12	6, 11	impbii 212	1 ⊢ (𝐴 = (𝐵 ∪ 𝐶) ↔ 𝐴 = (𝐶 ∪ 𝐵))

Syntax hints:   ↔ wb 209   = wceq 1538   ∪ cun 3879

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-un 3886  df-vd1 41276

This theorem is referenced by: (None)