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Theorem nfopd 3943
Description: Deduction version of bound-variable hypothesis builder nfop 3942. This shows how the deduction version of a not-free theorem such as nfop 3942 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 2528 . . 3  |-  F/_ x { z  |  A. x  z  e.  A }
2 nfaba1 2528 . . 3  |-  F/_ x { z  |  A. x  z  e.  B }
31, 2nfop 3942 . 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 2526 . . . . 5  |-  F/ x F/_ x A
7 nfnfc1 2526 . . . . 5  |-  F/ x F/_ x B
86, 7nfan 1836 . . . 4  |-  F/ x
( F/_ x A  /\  F/_ x B )
9 abidnf 3046 . . . . . 6  |-  ( F/_ x A  ->  { z  |  A. x  z  e.  A }  =  A )
109adantr 452 . . . . 5  |-  ( (
F/_ x A  /\  F/_ x B )  ->  { z  |  A. x  z  e.  A }  =  A )
11 abidnf 3046 . . . . . 6  |-  ( F/_ x B  ->  { z  |  A. x  z  e.  B }  =  B )
1211adantl 453 . . . . 5  |-  ( (
F/_ x A  /\  F/_ x B )  ->  { z  |  A. x  z  e.  B }  =  B )
1310, 12opeq12d 3934 . . . 4  |-  ( (
F/_ x A  /\  F/_ x B )  ->  <. { z  |  A. x  z  e.  A } ,  { z  |  A. x  z  e.  B } >.  =  <. A ,  B >. )
148, 13nfceqdf 2522 . . 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 643 . 2  |-  ( ph  ->  ( F/_ x <. { z  |  A. x  z  e.  A } ,  { z  |  A. x  z  e.  B } >. 
<-> 
F/_ x <. A ,  B >. ) )
163, 15mpbii 203 1  |-  ( ph  -> 
F/_ x <. A ,  B >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546    = wceq 1649    e. wcel 1717   {cab 2373   F/_wnfc 2510   <.cop 3760
This theorem is referenced by:  nfbrd  4196  dfid3  4440  nfovd  6042
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766
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