MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dvelim Unicode version

Theorem dvelim 2079
Description: This theorem can be used to eliminate a distinct variable restriction on and and replace it with the "distinctor" as an antecedent. normally has free and can be read ( ), and substitutes for and can be read ( ). We do not require that and be distinct: if they are not, the distinctor will become false (in multiple-element domains of discourse) and "protect" the consequent.

To obtain a closed-theorem form of this inference, prefix the hypotheses with A.xA. , conjoin them, and apply dvelimdf 2077.

Other variants of this theorem are dvelimh 2078 (with no distinct variable restrictions) and dvelimhw 1955 (that avoids ax-13 1999). (Contributed by NM, 23-Nov-1994.)

Hypotheses
Ref Expression
dvelim.1
dvelim.2
Assertion
Ref Expression
dvelim
Distinct variable group:   ,

Proof of Theorem dvelim
StepHypRef Expression
1 dvelim.1 . 2
2 ax-5 1704 . 2
3 dvelim.2 . 2
41, 2, 3dvelimh 2078 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  <->wb 184  A.wal 1393
This theorem is referenced by:  dvelimv  2080  axc14  2113  eujustALT  2285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1613  df-nf 1617
  Copyright terms: Public domain W3C validator