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Theorem equncomVD 39418
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 3791 is equncomVD 39418 without virtual deductions and was automatically derived from equncomVD 39418.
 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 39107 . . . 4 (   𝐴 = (𝐵𝐶)   ▶   𝐴 = (𝐵𝐶)   )
2 uncom 3790 . . . 4 (𝐵𝐶) = (𝐶𝐵)
3 eqeq1 2655 . . . . 5 (𝐴 = (𝐵𝐶) → (𝐴 = (𝐶𝐵) ↔ (𝐵𝐶) = (𝐶𝐵)))
43biimprd 238 . . . 4 (𝐴 = (𝐵𝐶) → ((𝐵𝐶) = (𝐶𝐵) → 𝐴 = (𝐶𝐵)))
51, 2, 4e10 39236 . . 3 (   𝐴 = (𝐵𝐶)   ▶   𝐴 = (𝐶𝐵)   )
65in1 39104 . 2 (𝐴 = (𝐵𝐶) → 𝐴 = (𝐶𝐵))
7 idn1 39107 . . . 4 (   𝐴 = (𝐶𝐵)   ▶   𝐴 = (𝐶𝐵)   )
8 eqeq2 2662 . . . . 5 ((𝐵𝐶) = (𝐶𝐵) → (𝐴 = (𝐵𝐶) ↔ 𝐴 = (𝐶𝐵)))
98biimprcd 240 . . . 4 (𝐴 = (𝐶𝐵) → ((𝐵𝐶) = (𝐶𝐵) → 𝐴 = (𝐵𝐶)))
107, 2, 9e10 39236 . . 3 (   𝐴 = (𝐶𝐵)   ▶   𝐴 = (𝐵𝐶)   )
1110in1 39104 . 2 (𝐴 = (𝐶𝐵) → 𝐴 = (𝐵𝐶))
126, 11impbii 199 1 (𝐴 = (𝐵𝐶) ↔ 𝐴 = (𝐶𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   = wceq 1523   ∪ cun 3605 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-un 3612  df-vd1 39103 This theorem is referenced by: (None)
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