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Theorem nffvd 5508
Description: Deduction version of bound-variable hypothesis builder nffv 5506. (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 2318 . . 3  |-  F/_ x { z  |  A. x  z  e.  F }
2 nfaba1 2318 . . 3  |-  F/_ x { z  |  A. x  z  e.  A }
31, 2nffv 5506 . 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 2315 . . . . 5  |-  F/ x F/_ x F
7 nfnfc1 2315 . . . . 5  |-  F/ x F/_ x A
86, 7nfan 1558 . . . 4  |-  F/ x
( F/_ x F  /\  F/_ x A )
9 abidnf 2898 . . . . . 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 2898 . . . . . 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 5503 . . . 4  |-  ( (
F/_ x F  /\  F/_ x A )  -> 
( { z  | 
A. x  z  e.  F } `  {
z  |  A. x  z  e.  A }
)  =  ( F `
 A ) )
148, 13nfceqdf 2311 . . 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 409 . 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 1346    = wceq 1348    e. wcel 2141   {cab 2156   F/_wnfc 2299   ` cfv 5198
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-fv 5206
This theorem is referenced by:  nfovd  5882  nfixpxy  6695
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