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Theorem equncomVD 39852
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 3954 is equncomVD 39852 without virtual deductions and was automatically derived from equncomVD 39852.
 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 = (𝐵𝐶) ↔ 𝐴 = (𝐶𝐵))

Proof of Theorem equncomVD
StepHypRef Expression
1 idn1 39548 . . . 4 (   𝐴 = (𝐵𝐶)   ▶   𝐴 = (𝐵𝐶)   )
2 uncom 3953 . . . 4 (𝐵𝐶) = (𝐶𝐵)
3 eqeq1 2801 . . . . 5 (𝐴 = (𝐵𝐶) → (𝐴 = (𝐶𝐵) ↔ (𝐵𝐶) = (𝐶𝐵)))
43biimprd 240 . . . 4 (𝐴 = (𝐵𝐶) → ((𝐵𝐶) = (𝐶𝐵) → 𝐴 = (𝐶𝐵)))
51, 2, 4e10 39677 . . 3 (   𝐴 = (𝐵𝐶)   ▶   𝐴 = (𝐶𝐵)   )
65in1 39545 . 2 (𝐴 = (𝐵𝐶) → 𝐴 = (𝐶𝐵))
7 idn1 39548 . . . 4 (   𝐴 = (𝐶𝐵)   ▶   𝐴 = (𝐶𝐵)   )
8 eqeq2 2808 . . . . 5 ((𝐵𝐶) = (𝐶𝐵) → (𝐴 = (𝐵𝐶) ↔ 𝐴 = (𝐶𝐵)))
98biimprcd 242 . . . 4 (𝐴 = (𝐶𝐵) → ((𝐵𝐶) = (𝐶𝐵) → 𝐴 = (𝐵𝐶)))
107, 2, 9e10 39677 . . 3 (   𝐴 = (𝐶𝐵)   ▶   𝐴 = (𝐵𝐶)   )
1110in1 39545 . 2 (𝐴 = (𝐶𝐵) → 𝐴 = (𝐵𝐶))
126, 11impbii 201 1 (𝐴 = (𝐵𝐶) ↔ 𝐴 = (𝐶𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   = wceq 1653   ∪ cun 3765 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2775 This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-v 3385  df-un 3772  df-vd1 39544 This theorem is referenced by: (None)
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