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Theorem nfopd 4196
Description: Deduction version of bound-variable hypothesis builder nfop 4195. This shows how the deduction version of a not-free theorem such as nfop 4195 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 2607 . . 3  |-  F/_ x { z  |  A. x  z  e.  A }
2 nfaba1 2607 . . 3  |-  F/_ x { z  |  A. x  z  e.  B }
31, 2nfop 4195 . 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 2605 . . . . 5  |-  F/ x F/_ x A
7 nfnfc1 2605 . . . . 5  |-  F/ x F/_ x B
86, 7nfan 2021 . . . 4  |-  F/ x
( F/_ x A  /\  F/_ x B )
9 abidnf 3218 . . . . . 6  |-  ( F/_ x A  ->  { z  |  A. x  z  e.  A }  =  A )
109adantr 471 . . . . 5  |-  ( (
F/_ x A  /\  F/_ x B )  ->  { z  |  A. x  z  e.  A }  =  A )
11 abidnf 3218 . . . . . 6  |-  ( F/_ x B  ->  { z  |  A. x  z  e.  B }  =  B )
1211adantl 472 . . . . 5  |-  ( (
F/_ x A  /\  F/_ x B )  ->  { z  |  A. x  z  e.  B }  =  B )
1310, 12opeq12d 4187 . . . 4  |-  ( (
F/_ x A  /\  F/_ x B )  ->  <. { z  |  A. x  z  e.  A } ,  { z  |  A. x  z  e.  B } >.  =  <. A ,  B >. )
148, 13nfceqdf 2598 . . 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 671 . 2  |-  ( ph  ->  ( F/_ x <. { z  |  A. x  z  e.  A } ,  { z  |  A. x  z  e.  B } >. 
<-> 
F/_ x <. A ,  B >. ) )
163, 15mpbii 216 1  |-  ( ph  -> 
F/_ x <. A ,  B >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375   A.wal 1452    = wceq 1454    e. wcel 1897   {cab 2447   F/_wnfc 2589   <.cop 3985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-rab 2757  df-v 3058  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986
This theorem is referenced by:  nfbrd  4459  dfid3  4768  nfovd  6339
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