NFE Home New Foundations Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  NFE Home  >  Th. List  >  neeq2 GIF version

Theorem neeq2 2525
Description: Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
Assertion
Ref Expression
neeq2 (A = B → (CACB))

Proof of Theorem neeq2
StepHypRef Expression
1 eqeq2 2362 . . 3 (A = B → (C = AC = B))
21notbid 285 . 2 (A = B → (¬ C = A ↔ ¬ C = B))
3 df-ne 2518 . 2 (CA ↔ ¬ C = A)
4 df-ne 2518 . 2 (CB ↔ ¬ C = B)
52, 3, 43bitr4g 279 1 (A = B → (CACB))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 176   = wceq 1642  wne 2516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-ex 1542  df-cleq 2346  df-ne 2518
This theorem is referenced by:  neeq2i  2527  neeq2d  2530  psseq2  3357  nfunv  5138  enprmapc  6083  ce2  6192
  Copyright terms: Public domain W3C validator