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Theorem dedhb 3064
Description: A deduction theorem for converting the inference  |-  F/_ x A =>  |-  ph into a closed theorem. Use nfa1 1802 and nfab 2544 to eliminate the hypothesis of the substitution instance  ps of the inference. For converting the inference form into a deduction form, abidnf 3063 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 3063 . . . 4  |-  ( F/_ x A  ->  { z  |  A. x  z  e.  A }  =  A )
32eqcomd 2409 . . 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 16 . 2  |-  ( F/_ x A  ->  ( ph  <->  ps ) )
61, 5mpbiri 225 1  |-  ( F/_ x A  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1546    = wceq 1649    e. wcel 1721   {cab 2390   F/_wnfc 2527
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  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529
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