MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfrald Unicode version

Theorem nfrald 2700
Description: Deduction version of nfral 2702. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.)
Hypotheses
Ref Expression
nfrald.2  |-  F/ y
ph
nfrald.3  |-  ( ph  -> 
F/_ x A )
nfrald.4  |-  ( ph  ->  F/ x ps )
Assertion
Ref Expression
nfrald  |-  ( ph  ->  F/ x A. y  e.  A  ps )

Proof of Theorem nfrald
StepHypRef Expression
1 df-ral 2654 . 2  |-  ( A. y  e.  A  ps  <->  A. y ( y  e.  A  ->  ps )
)
2 nfrald.2 . . 3  |-  F/ y
ph
3 nfcvf 2545 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  F/_ x y )
43adantl 453 . . . . 5  |-  ( (
ph  /\  -.  A. x  x  =  y )  -> 
F/_ x y )
5 nfrald.3 . . . . . 6  |-  ( ph  -> 
F/_ x A )
65adantr 452 . . . . 5  |-  ( (
ph  /\  -.  A. x  x  =  y )  -> 
F/_ x A )
74, 6nfeld 2538 . . . 4  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x  y  e.  A )
8 nfrald.4 . . . . 5  |-  ( ph  ->  F/ x ps )
98adantr 452 . . . 4  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ps )
107, 9nfimd 1817 . . 3  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ( y  e.  A  ->  ps ) )
112, 10nfald2 2011 . 2  |-  ( ph  ->  F/ x A. y
( y  e.  A  ->  ps ) )
121, 11nfxfrd 1577 1  |-  ( ph  ->  F/ x A. y  e.  A  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359   A.wal 1546   F/wnf 1550    e. wcel 1717   F/_wnfc 2510   A.wral 2649
This theorem is referenced by:  nfrexd  2701  nfral  2702  riotasvdOLD  6529
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654
  Copyright terms: Public domain W3C validator