 New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  eqeq12 GIF version

Theorem eqeq12 2365
 Description: Equality relationship among 4 classes. (Contributed by NM, 3-Aug-1994.)
Assertion
Ref Expression
eqeq12 ((A = B C = D) → (A = CB = D))

Proof of Theorem eqeq12
StepHypRef Expression
1 eqeq1 2359 . 2 (A = B → (A = CB = C))
2 eqeq2 2362 . 2 (C = D → (B = CB = D))
31, 2sylan9bb 680 1 ((A = B C = D) → (A = CB = D))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346 This theorem is referenced by:  eqeq12i  2366  eqeq12d  2367  eqeqan12d  2368  dfpw12  4301  fununiq  5517  fntxp  5804  pw1fnf1o  5855  fundmen  6043  ncdisjeq  6148  peano4nc  6150  sbth  6206  tc11  6228  fnfrec  6320
 Copyright terms: Public domain W3C validator