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Theorem nfopd 3634
Description: Deduction version of bound-variable hypothesis builder nfop 3633. This shows how the deduction version of a not-free theorem such as nfop 3633 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 2234 . . 3  |-  F/_ x { z  |  A. x  z  e.  A }
2 nfaba1 2234 . . 3  |-  F/_ x { z  |  A. x  z  e.  B }
31, 2nfop 3633 . 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 2231 . . . . 5  |-  F/ x F/_ x A
7 nfnfc1 2231 . . . . 5  |-  F/ x F/_ x B
86, 7nfan 1502 . . . 4  |-  F/ x
( F/_ x A  /\  F/_ x B )
9 abidnf 2781 . . . . . 6  |-  ( F/_ x A  ->  { z  |  A. x  z  e.  A }  =  A )
109adantr 270 . . . . 5  |-  ( (
F/_ x A  /\  F/_ x B )  ->  { z  |  A. x  z  e.  A }  =  A )
11 abidnf 2781 . . . . . 6  |-  ( F/_ x B  ->  { z  |  A. x  z  e.  B }  =  B )
1211adantl 271 . . . . 5  |-  ( (
F/_ x A  /\  F/_ x B )  ->  { z  |  A. x  z  e.  B }  =  B )
1310, 12opeq12d 3625 . . . 4  |-  ( (
F/_ x A  /\  F/_ x B )  ->  <. { z  |  A. x  z  e.  A } ,  { z  |  A. x  z  e.  B } >.  =  <. A ,  B >. )
148, 13nfceqdf 2227 . . 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 403 . 2  |-  ( ph  ->  ( F/_ x <. { z  |  A. x  z  e.  A } ,  { z  |  A. x  z  e.  B } >. 
<-> 
F/_ x <. A ,  B >. ) )
163, 15mpbii 146 1  |-  ( ph  -> 
F/_ x <. A ,  B >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1287    = wceq 1289    e. wcel 1438   {cab 2074   F/_wnfc 2215   <.cop 3444
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-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-sn 3447  df-pr 3448  df-op 3450
This theorem is referenced by:  nfbrd  3880  nfovd  5660
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