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Theorem nffvd 5534
Description: Deduction version of bound-variable hypothesis builder nffv 5532. (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 2424 . . 3  |-  F/_ x { z  |  A. x  z  e.  F }
2 nfaba1 2424 . . 3  |-  F/_ x { z  |  A. x  z  e.  A }
31, 2nffv 5532 . 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 2422 . . . . 5  |-  F/ x F/_ x F
7 nfnfc1 2422 . . . . 5  |-  F/ x F/_ x A
86, 7nfan 1771 . . . 4  |-  F/ x
( F/_ x F  /\  F/_ x A )
9 abidnf 2934 . . . . . 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 2934 . . . . . 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 5531 . . . 4  |-  ( (
F/_ x F  /\  F/_ x A )  -> 
( { z  | 
A. x  z  e.  F } `  {
z  |  A. x  z  e.  A }
)  =  ( F `
 A ) )
148, 13nfceqdf 2418 . . 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 1527    = wceq 1623    e. wcel 1684   {cab 2269   F/_wnfc 2406   ` cfv 5255
This theorem is referenced by:  nfovd  5880  nfriotad  6313  nfixp  6835
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263
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