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Theorem dvelim 2052
 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 , conjoin them, and apply dvelimdf 2050. Other variants of this theorem are dvelimh 2051 (with no distinct variable restrictions) and dvelimhw 1902 (that avoids ax-13 1968). (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-5 1680 . 2
3 dvelim.2 . 2
41, 2, 3dvelimh 2051 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 184  wal 1377 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968 This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1597  df-nf 1600 This theorem is referenced by:  dvelimv  2053  axc14  2086  eujustALT  2278
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