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Theorem equncomVD 41494
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 4116 is equncomVD 41494 without virtual deductions and was automatically derived from equncomVD 41494.
 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 41200 . . . 4 (   𝐴 = (𝐵𝐶)   ▶   𝐴 = (𝐵𝐶)   )
2 uncom 4115 . . . 4 (𝐵𝐶) = (𝐶𝐵)
3 eqeq1 2828 . . . . 5 (𝐴 = (𝐵𝐶) → (𝐴 = (𝐶𝐵) ↔ (𝐵𝐶) = (𝐶𝐵)))
43biimprd 251 . . . 4 (𝐴 = (𝐵𝐶) → ((𝐵𝐶) = (𝐶𝐵) → 𝐴 = (𝐶𝐵)))
51, 2, 4e10 41320 . . 3 (   𝐴 = (𝐵𝐶)   ▶   𝐴 = (𝐶𝐵)   )
65in1 41197 . 2 (𝐴 = (𝐵𝐶) → 𝐴 = (𝐶𝐵))
7 idn1 41200 . . . 4 (   𝐴 = (𝐶𝐵)   ▶   𝐴 = (𝐶𝐵)   )
8 eqeq2 2836 . . . . 5 ((𝐵𝐶) = (𝐶𝐵) → (𝐴 = (𝐵𝐶) ↔ 𝐴 = (𝐶𝐵)))
98biimprcd 253 . . . 4 (𝐴 = (𝐶𝐵) → ((𝐵𝐶) = (𝐶𝐵) → 𝐴 = (𝐵𝐶)))
107, 2, 9e10 41320 . . 3 (   𝐴 = (𝐶𝐵)   ▶   𝐴 = (𝐵𝐶)   )
1110in1 41197 . 2 (𝐴 = (𝐶𝐵) → 𝐴 = (𝐵𝐶))
126, 11impbii 212 1 (𝐴 = (𝐵𝐶) ↔ 𝐴 = (𝐶𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538   ∪ cun 3917 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3482  df-un 3924  df-vd1 41196 This theorem is referenced by: (None)
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