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

Theorem nfopd 4220
Description: Deduction version of bound-variable hypothesis builder nfop 4219. This shows how the deduction version of a not-free theorem such as nfop 4219 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 2621 . . 3  |-  F/_ x { z  |  A. x  z  e.  A }
2 nfaba1 2621 . . 3  |-  F/_ x { z  |  A. x  z  e.  B }
31, 2nfop 4219 . 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 2619 . . . . 5  |-  F/ x F/_ x A
7 nfnfc1 2619 . . . . 5  |-  F/ x F/_ x B
86, 7nfan 1933 . . . 4  |-  F/ x
( F/_ x A  /\  F/_ x B )
9 abidnf 3265 . . . . . 6  |-  ( F/_ x A  ->  { z  |  A. x  z  e.  A }  =  A )
109adantr 463 . . . . 5  |-  ( (
F/_ x A  /\  F/_ x B )  ->  { z  |  A. x  z  e.  A }  =  A )
11 abidnf 3265 . . . . . 6  |-  ( F/_ x B  ->  { z  |  A. x  z  e.  B }  =  B )
1211adantl 464 . . . . 5  |-  ( (
F/_ x A  /\  F/_ x B )  ->  { z  |  A. x  z  e.  B }  =  B )
1310, 12opeq12d 4211 . . . 4  |-  ( (
F/_ x A  /\  F/_ x B )  ->  <. { z  |  A. x  z  e.  A } ,  { z  |  A. x  z  e.  B } >.  =  <. A ,  B >. )
148, 13nfceqdf 2611 . . 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 659 . 2  |-  ( ph  ->  ( F/_ x <. { z  |  A. x  z  e.  A } ,  { z  |  A. x  z  e.  B } >. 
<-> 
F/_ x <. A ,  B >. ) )
163, 15mpbii 211 1  |-  ( ph  -> 
F/_ x <. A ,  B >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1396    = wceq 1398    e. wcel 1823   {cab 2439   F/_wnfc 2602   <.cop 4022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023
This theorem is referenced by:  nfbrd  4482  dfid3  4785  nfovd  6295
  Copyright terms: Public domain W3C validator