Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  equncomVD Structured version   Visualization version   GIF version

Theorem equncomVD 44865
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 4168 is equncomVD 44865 without virtual deductions and was automatically derived from equncomVD 44865.
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 44571 . . . 4 (   𝐴 = (𝐵𝐶)   ▶   𝐴 = (𝐵𝐶)   )
2 uncom 4167 . . . 4 (𝐵𝐶) = (𝐶𝐵)
3 eqeq1 2738 . . . . 5 (𝐴 = (𝐵𝐶) → (𝐴 = (𝐶𝐵) ↔ (𝐵𝐶) = (𝐶𝐵)))
43biimprd 248 . . . 4 (𝐴 = (𝐵𝐶) → ((𝐵𝐶) = (𝐶𝐵) → 𝐴 = (𝐶𝐵)))
51, 2, 4e10 44691 . . 3 (   𝐴 = (𝐵𝐶)   ▶   𝐴 = (𝐶𝐵)   )
65in1 44568 . 2 (𝐴 = (𝐵𝐶) → 𝐴 = (𝐶𝐵))
7 idn1 44571 . . . 4 (   𝐴 = (𝐶𝐵)   ▶   𝐴 = (𝐶𝐵)   )
8 eqeq2 2746 . . . . 5 ((𝐵𝐶) = (𝐶𝐵) → (𝐴 = (𝐵𝐶) ↔ 𝐴 = (𝐶𝐵)))
98biimprcd 250 . . . 4 (𝐴 = (𝐶𝐵) → ((𝐵𝐶) = (𝐶𝐵) → 𝐴 = (𝐵𝐶)))
107, 2, 9e10 44691 . . 3 (   𝐴 = (𝐶𝐵)   ▶   𝐴 = (𝐵𝐶)   )
1110in1 44568 . 2 (𝐴 = (𝐶𝐵) → 𝐴 = (𝐵𝐶))
126, 11impbii 209 1 (𝐴 = (𝐵𝐶) ↔ 𝐴 = (𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1536  cun 3960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-v 3479  df-un 3967  df-vd1 44567
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator