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Theorem dedhb 2853
Description: A deduction theorem for converting the inference  |-  F/_ x A =>  |-  ph into a closed theorem. Use nfa1 1521 and nfab 2286 to eliminate the hypothesis of the substitution instance  ps of the inference. For converting the inference form into a deduction form, abidnf 2852 is useful. (Contributed by NM, 8-Dec-2006.)
Hypotheses
Ref Expression
dedhb.1  |-  ( A  =  { z  | 
A. x  z  e.  A }  ->  ( ph 
<->  ps ) )
dedhb.2  |-  ps
Assertion
Ref Expression
dedhb  |-  ( F/_ x A  ->  ph )
Distinct variable groups:    x, z    z, A
Allowed substitution hints:    ph( x, z)    ps( x, z)    A( x)

Proof of Theorem dedhb
StepHypRef Expression
1 dedhb.2 . 2  |-  ps
2 abidnf 2852 . . . 4  |-  ( F/_ x A  ->  { z  |  A. x  z  e.  A }  =  A )
32eqcomd 2145 . . 3  |-  ( F/_ x A  ->  A  =  { z  |  A. x  z  e.  A } )
4 dedhb.1 . . 3  |-  ( A  =  { z  | 
A. x  z  e.  A }  ->  ( ph 
<->  ps ) )
53, 4syl 14 . 2  |-  ( F/_ x A  ->  ( ph  <->  ps ) )
61, 5mpbiri 167 1  |-  ( F/_ x A  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1329    = wceq 1331    e. wcel 1480   {cab 2125   F/_wnfc 2268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270
This theorem is referenced by: (None)
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