MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dedhb Unicode version

Theorem dedhb 2948
Description: A deduction theorem for converting the inference  |-  F/_ x A =>  |-  ph into a closed theorem. Use nfa1 1768 and nfab 2436 to eliminate the hypothesis of the substitution instance  ps of the inference. For converting the inference form into a deduction form, abidnf 2947 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 2947 . . . 4  |-  ( F/_ x A  ->  { z  |  A. x  z  e.  A }  =  A )
32eqcomd 2301 . . 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 15 . 2  |-  ( F/_ x A  ->  ( ph  <->  ps ) )
61, 5mpbiri 224 1  |-  ( F/_ x A  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1530    = wceq 1632    e. wcel 1696   {cab 2282   F/_wnfc 2419
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421
  Copyright terms: Public domain W3C validator