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Theorem dedhb 2772
Description: A deduction theorem for converting the inference  |-  F/_ x A =>  |-  ph into a closed theorem. Use nfa1 1475 and nfab 2227 to eliminate the hypothesis of the substitution instance  ps of the inference. For converting the inference form into a deduction form, abidnf 2771 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 2771 . . . 4  |-  ( F/_ x A  ->  { z  |  A. x  z  e.  A }  =  A )
32eqcomd 2088 . . 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 166 1  |-  ( F/_ x A  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1283    = wceq 1285    e. wcel 1434   {cab 2069   F/_wnfc 2210
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212
This theorem is referenced by: (None)
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