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Theorem dvelim 2069
Description: This theorem can be used to eliminate a distinct variable restriction on  x and  z and replace it with the "distinctor"  -.  A. x x  =  y as an antecedent.  ph normally has  z free and can be read  ph ( z ), and  ps substitutes  y for  z and can be read  ph ( y ). We don't require that 
x and  y 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  A. x A. z, 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  |-  ( ph  ->  A. x ph )
dvelim.2  |-  ( z  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
dvelim  |-  ( -. 
A. x  x  =  y  ->  ( ps  ->  A. x ps )
)
Distinct variable group:    ps, z
Allowed substitution hints:    ph( x, y, z)    ps( x, y)

Proof of Theorem dvelim
StepHypRef Expression
1 dvelim.1 . 2  |-  ( ph  ->  A. x ph )
2 ax-17 1626 . 2  |-  ( ps 
->  A. z ps )
3 dvelim.2 . 2  |-  ( z  =  y  ->  ( ph 
<->  ps ) )
41, 2, 3dvelimh 2067 1  |-  ( -. 
A. x  x  =  y  ->  ( ps  ->  A. x ps )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177   A.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
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