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

Theorem dvelim 2069
 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 don't require that and be distinct: if they aren't, 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 , conjoin them, and apply dvelimdf 2066. Other variants of this theorem are dvelimh 2067 (with no distinct variable restrictions), dvelimhw 1876 (that avoids ax-12 1950), and dvelimALT 2209 (that avoids ax-10 2216). (Contributed by NM, 23-Nov-1994.)
Hypotheses
Ref Expression
dvelim.1
dvelim.2
Assertion
Ref Expression
dvelim
Distinct variable group:   ,
Allowed substitution hints:   (,,)   (,)

Proof of Theorem dvelim
StepHypRef Expression
1 dvelim.1 . 2
2 ax-17 1626 . 2
3 dvelim.2 . 2
41, 2, 3dvelimh 2067 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177  wal 1549 This theorem is referenced by:  dvelimv  2070  ax15  2101  eujustALT  2283 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950 This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554
 Copyright terms: Public domain W3C validator