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Theorem nffvd 5401
Description: Deduction version of bound-variable hypothesis builder nffv 5399. (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 2264 . . 3  |-  F/_ x { z  |  A. x  z  e.  F }
2 nfaba1 2264 . . 3  |-  F/_ x { z  |  A. x  z  e.  A }
31, 2nffv 5399 . 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 2261 . . . . 5  |-  F/ x F/_ x F
7 nfnfc1 2261 . . . . 5  |-  F/ x F/_ x A
86, 7nfan 1529 . . . 4  |-  F/ x
( F/_ x F  /\  F/_ x A )
9 abidnf 2825 . . . . . 6  |-  ( F/_ x F  ->  { z  |  A. x  z  e.  F }  =  F )
109adantr 274 . . . . 5  |-  ( (
F/_ x F  /\  F/_ x A )  ->  { z  |  A. x  z  e.  F }  =  F )
11 abidnf 2825 . . . . . 6  |-  ( F/_ x A  ->  { z  |  A. x  z  e.  A }  =  A )
1211adantl 275 . . . . 5  |-  ( (
F/_ x F  /\  F/_ x A )  ->  { z  |  A. x  z  e.  A }  =  A )
1310, 12fveq12d 5396 . . . 4  |-  ( (
F/_ x F  /\  F/_ x A )  -> 
( { z  | 
A. x  z  e.  F } `  {
z  |  A. x  z  e.  A }
)  =  ( F `
 A ) )
148, 13nfceqdf 2257 . . 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 408 . 2  |-  ( ph  ->  ( F/_ x ( { z  |  A. x  z  e.  F } `  { z  |  A. x  z  e.  A } )  <->  F/_ x ( F `  A ) ) )
163, 15mpbii 147 1  |-  ( ph  -> 
F/_ x ( F `
 A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1314    = wceq 1316    e. wcel 1465   {cab 2103   F/_wnfc 2245   ` cfv 5093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-iota 5058  df-fv 5101
This theorem is referenced by:  nfovd  5768  nfixpxy  6579
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