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Theorem dvelim 2066
 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 2131. Other variants of this theorem are dvelimh 2015 (with no distinct variable restrictions), dvelimhw 1872 (that avoids ax-12 1946), and dvelimALT 2183 (that avoids ax-10 2190). (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 1623 . 2
3 dvelim.2 . 2
41, 2, 3dvelimh 2015 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177  wal 1546 This theorem is referenced by:  ax15  2070  eujustALT  2257 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946 This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551
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