Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  uneq12 GIF version

Theorem uneq12 3120
 Description: Equality theorem for union of two classes. (Contributed by NM, 29-Mar-1998.)
Assertion
Ref Expression
uneq12 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))

Proof of Theorem uneq12
StepHypRef Expression
1 uneq1 3118 . 2 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
2 uneq2 3119 . 2 (𝐶 = 𝐷 → (𝐵𝐶) = (𝐵𝐷))
31, 2sylan9eq 2108 1 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   = wceq 1259   ∪ cun 2943 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950 This theorem is referenced by:  uneq12i  3123  uneq12d  3126  un00  3291  opthprc  4419  dmpropg  4821  unixpm  4881  fntpg  4983  fnun  5033  resasplitss  5097
 Copyright terms: Public domain W3C validator