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Theorem equncomVD 43140
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 4114 is equncomVD 43140 without virtual deductions and was automatically derived from equncomVD 43140.
1:: (   𝐴 = (𝐵𝐶)   ▶   𝐴 = (𝐵𝐶)   )
2:: (𝐵𝐶) = (𝐶𝐵)
3:1,2: (   𝐴 = (𝐵𝐶)   ▶   𝐴 = (𝐶𝐵)   )
4:3: (𝐴 = (𝐵𝐶) → 𝐴 = (𝐶𝐵))
5:: (   𝐴 = (𝐶𝐵)   ▶   𝐴 = (𝐶𝐵)   )
6:5,2: (   𝐴 = (𝐶𝐵)   ▶   𝐴 = (𝐵𝐶)   )
7:6: (𝐴 = (𝐶𝐵) → 𝐴 = (𝐵𝐶))
8:4,7: (𝐴 = (𝐵𝐶) ↔ 𝐴 = (𝐶𝐵))
(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 42846 . . . 4 (   𝐴 = (𝐵𝐶)   ▶   𝐴 = (𝐵𝐶)   )
2 uncom 4113 . . . 4 (𝐵𝐶) = (𝐶𝐵)
3 eqeq1 2740 . . . . 5 (𝐴 = (𝐵𝐶) → (𝐴 = (𝐶𝐵) ↔ (𝐵𝐶) = (𝐶𝐵)))
43biimprd 247 . . . 4 (𝐴 = (𝐵𝐶) → ((𝐵𝐶) = (𝐶𝐵) → 𝐴 = (𝐶𝐵)))
51, 2, 4e10 42966 . . 3 (   𝐴 = (𝐵𝐶)   ▶   𝐴 = (𝐶𝐵)   )
65in1 42843 . 2 (𝐴 = (𝐵𝐶) → 𝐴 = (𝐶𝐵))
7 idn1 42846 . . . 4 (   𝐴 = (𝐶𝐵)   ▶   𝐴 = (𝐶𝐵)   )
8 eqeq2 2748 . . . . 5 ((𝐵𝐶) = (𝐶𝐵) → (𝐴 = (𝐵𝐶) ↔ 𝐴 = (𝐶𝐵)))
98biimprcd 249 . . . 4 (𝐴 = (𝐶𝐵) → ((𝐵𝐶) = (𝐶𝐵) → 𝐴 = (𝐵𝐶)))
107, 2, 9e10 42966 . . 3 (   𝐴 = (𝐶𝐵)   ▶   𝐴 = (𝐵𝐶)   )
1110in1 42843 . 2 (𝐴 = (𝐶𝐵) → 𝐴 = (𝐵𝐶))
126, 11impbii 208 1 (𝐴 = (𝐵𝐶) ↔ 𝐴 = (𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1541  cun 3908
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3447  df-un 3915  df-vd1 42842
This theorem is referenced by: (None)
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