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Theorem nfopd 3821
Description: Deduction version of bound-variable hypothesis builder nfop 3820. This shows how the deduction version of a not-free theorem such as nfop 3820 can be created from the corresponding not-free inference theorem. (Contributed by NM, 4-Feb-2008.)
Hypotheses
Ref Expression
nfopd.2  |-  ( ph  -> 
F/_ x A )
nfopd.3  |-  ( ph  -> 
F/_ x B )
Assertion
Ref Expression
nfopd  |-  ( ph  -> 
F/_ x <. A ,  B >. )

Proof of Theorem nfopd
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 nfaba1 2342 . . 3  |-  F/_ x { z  |  A. x  z  e.  A }
2 nfaba1 2342 . . 3  |-  F/_ x { z  |  A. x  z  e.  B }
31, 2nfop 3820 . 2  |-  F/_ x <. { z  |  A. x  z  e.  A } ,  { z  |  A. x  z  e.  B } >.
4 nfopd.2 . . 3  |-  ( ph  -> 
F/_ x A )
5 nfopd.3 . . 3  |-  ( ph  -> 
F/_ x B )
6 nfnfc1 2339 . . . . 5  |-  F/ x F/_ x A
7 nfnfc1 2339 . . . . 5  |-  F/ x F/_ x B
86, 7nfan 1576 . . . 4  |-  F/ x
( F/_ x A  /\  F/_ x B )
9 abidnf 2928 . . . . . 6  |-  ( F/_ x A  ->  { z  |  A. x  z  e.  A }  =  A )
109adantr 276 . . . . 5  |-  ( (
F/_ x A  /\  F/_ x B )  ->  { z  |  A. x  z  e.  A }  =  A )
11 abidnf 2928 . . . . . 6  |-  ( F/_ x B  ->  { z  |  A. x  z  e.  B }  =  B )
1211adantl 277 . . . . 5  |-  ( (
F/_ x A  /\  F/_ x B )  ->  { z  |  A. x  z  e.  B }  =  B )
1310, 12opeq12d 3812 . . . 4  |-  ( (
F/_ x A  /\  F/_ x B )  ->  <. { z  |  A. x  z  e.  A } ,  { z  |  A. x  z  e.  B } >.  =  <. A ,  B >. )
148, 13nfceqdf 2335 . . 3  |-  ( (
F/_ x A  /\  F/_ x B )  -> 
( F/_ x <. { z  |  A. x  z  e.  A } ,  { z  |  A. x  z  e.  B } >. 
<-> 
F/_ x <. A ,  B >. ) )
154, 5, 14syl2anc 411 . 2  |-  ( ph  ->  ( F/_ x <. { z  |  A. x  z  e.  A } ,  { z  |  A. x  z  e.  B } >. 
<-> 
F/_ x <. A ,  B >. ) )
163, 15mpbii 148 1  |-  ( ph  -> 
F/_ x <. A ,  B >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105   A.wal 1362    = wceq 1364    e. wcel 2164   {cab 2179   F/_wnfc 2323   <.cop 3621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627
This theorem is referenced by:  nfbrd  4074  nfovd  5943
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