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Theorem nffvd 5740
Description: Deduction version of bound-variable hypothesis builder nffv 5738. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
nffvd.2  |-  ( ph  -> 
F/_ x F )
nffvd.3  |-  ( ph  -> 
F/_ x A )
Assertion
Ref Expression
nffvd  |-  ( ph  -> 
F/_ x ( F `
 A ) )

Proof of Theorem nffvd
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 nfaba1 2579 . . 3  |-  F/_ x { z  |  A. x  z  e.  F }
2 nfaba1 2579 . . 3  |-  F/_ x { z  |  A. x  z  e.  A }
31, 2nffv 5738 . 2  |-  F/_ x
( { z  | 
A. x  z  e.  F } `  {
z  |  A. x  z  e.  A }
)
4 nffvd.2 . . 3  |-  ( ph  -> 
F/_ x F )
5 nffvd.3 . . 3  |-  ( ph  -> 
F/_ x A )
6 nfnfc1 2577 . . . . 5  |-  F/ x F/_ x F
7 nfnfc1 2577 . . . . 5  |-  F/ x F/_ x A
86, 7nfan 1847 . . . 4  |-  F/ x
( F/_ x F  /\  F/_ x A )
9 abidnf 3105 . . . . . 6  |-  ( F/_ x F  ->  { z  |  A. x  z  e.  F }  =  F )
109adantr 453 . . . . 5  |-  ( (
F/_ x F  /\  F/_ x A )  ->  { z  |  A. x  z  e.  F }  =  F )
11 abidnf 3105 . . . . . 6  |-  ( F/_ x A  ->  { z  |  A. x  z  e.  A }  =  A )
1211adantl 454 . . . . 5  |-  ( (
F/_ x F  /\  F/_ x A )  ->  { z  |  A. x  z  e.  A }  =  A )
1310, 12fveq12d 5737 . . . 4  |-  ( (
F/_ x F  /\  F/_ x A )  -> 
( { z  | 
A. x  z  e.  F } `  {
z  |  A. x  z  e.  A }
)  =  ( F `
 A ) )
148, 13nfceqdf 2573 . . 3  |-  ( (
F/_ x F  /\  F/_ x A )  -> 
( F/_ x ( { z  |  A. x  z  e.  F } `  { z  |  A. x  z  e.  A } )  <->  F/_ x ( F `  A ) ) )
154, 5, 14syl2anc 644 . 2  |-  ( ph  ->  ( F/_ x ( { z  |  A. x  z  e.  F } `  { z  |  A. x  z  e.  A } )  <->  F/_ x ( F `  A ) ) )
163, 15mpbii 204 1  |-  ( ph  -> 
F/_ x ( F `
 A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   A.wal 1550    = wceq 1653    e. wcel 1726   {cab 2424   F/_wnfc 2561   ` cfv 5457
This theorem is referenced by:  nfovd  6106  nfriotad  6561  nfixp  7084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-iota 5421  df-fv 5465
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