MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfald2 Unicode version

Theorem nfald2 2016
Description: Variation on nfald 1864 which adds the hypothesis that  x and  y are distinct in the inner subproof. (Contributed by Mario Carneiro, 8-Oct-2016.)
Hypotheses
Ref Expression
nfald2.1  |-  F/ y
ph
nfald2.2  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ps )
Assertion
Ref Expression
nfald2  |-  ( ph  ->  F/ x A. y ps )

Proof of Theorem nfald2
StepHypRef Expression
1 nfald2.1 . . . . 5  |-  F/ y
ph
2 nfnae 2005 . . . . 5  |-  F/ y  -.  A. x  x  =  y
31, 2nfan 1836 . . . 4  |-  F/ y ( ph  /\  -.  A. x  x  =  y )
4 nfald2.2 . . . 4  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ps )
53, 4nfald 1864 . . 3  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x A. y ps )
65ex 424 . 2  |-  ( ph  ->  ( -.  A. x  x  =  y  ->  F/ x A. y ps ) )
7 nfa1 1796 . . 3  |-  F/ y A. y ps
8 biidd 229 . . . 4  |-  ( A. x  x  =  y  ->  ( A. y ps  <->  A. y ps ) )
98drnf1 2013 . . 3  |-  ( A. x  x  =  y  ->  ( F/ x A. y ps  <->  F/ y A. y ps ) )
107, 9mpbiri 225 . 2  |-  ( A. x  x  =  y  ->  F/ x A. y ps )
116, 10pm2.61d2 154 1  |-  ( ph  ->  F/ x A. y ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359   A.wal 1546   F/wnf 1550
This theorem is referenced by:  nfexd2  2017  dvelimf  2036  nfeud2  2252  nfrald  2702  nfiotad  5363  nfixp  7019
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
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551
  Copyright terms: Public domain W3C validator