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Theorem nffvd 5550
Description: Deduction version of bound-variable hypothesis builder nffv 5548. (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 2437 . . 3  |-  F/_ x { z  |  A. x  z  e.  F }
2 nfaba1 2437 . . 3  |-  F/_ x { z  |  A. x  z  e.  A }
31, 2nffv 5548 . 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 2435 . . . . 5  |-  F/ x F/_ x F
7 nfnfc1 2435 . . . . 5  |-  F/ x F/_ x A
86, 7nfan 1783 . . . 4  |-  F/ x
( F/_ x F  /\  F/_ x A )
9 abidnf 2947 . . . . . 6  |-  ( F/_ x F  ->  { z  |  A. x  z  e.  F }  =  F )
109adantr 451 . . . . 5  |-  ( (
F/_ x F  /\  F/_ x A )  ->  { z  |  A. x  z  e.  F }  =  F )
11 abidnf 2947 . . . . . 6  |-  ( F/_ x A  ->  { z  |  A. x  z  e.  A }  =  A )
1211adantl 452 . . . . 5  |-  ( (
F/_ x F  /\  F/_ x A )  ->  { z  |  A. x  z  e.  A }  =  A )
1310, 12fveq12d 5547 . . . 4  |-  ( (
F/_ x F  /\  F/_ x A )  -> 
( { z  | 
A. x  z  e.  F } `  {
z  |  A. x  z  e.  A }
)  =  ( F `
 A ) )
148, 13nfceqdf 2431 . . 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 642 . 2  |-  ( ph  ->  ( F/_ x ( { z  |  A. x  z  e.  F } `  { z  |  A. x  z  e.  A } )  <->  F/_ x ( F `  A ) ) )
163, 15mpbii 202 1  |-  ( ph  -> 
F/_ x ( F `
 A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530    = wceq 1632    e. wcel 1696   {cab 2282   F/_wnfc 2419   ` cfv 5271
This theorem is referenced by:  nfovd  5896  nfriotad  6329  nfixp  6851
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279
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