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

Theorem nfald2 2030
Description: Variation on nfald 1886 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 2015 . . . . 5  |-  F/ y  -.  A. x  x  =  y
31, 2nfan 1863 . . . 4  |-  F/ y ( ph  /\  -.  A. x  x  =  y )
4 nfald2.2 . . . 4  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ps )
53, 4nfald 1886 . . 3  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x A. y ps )
65ex 434 . 2  |-  ( ph  ->  ( -.  A. x  x  =  y  ->  F/ x A. y ps ) )
7 nfa1 1833 . . 3  |-  F/ y A. y ps
8 biidd 237 . . . 4  |-  ( A. x  x  =  y  ->  ( A. y ps  <->  A. y ps ) )
98drnf1 2028 . . 3  |-  ( A. x  x  =  y  ->  ( F/ x A. y ps  <->  F/ y A. y ps ) )
107, 9mpbiri 233 . 2  |-  ( A. x  x  =  y  ->  F/ x A. y ps )
116, 10pm2.61d2 160 1  |-  ( ph  ->  F/ x A. y ps )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369   A.wal 1368   F/wnf 1590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-nf 1591
This theorem is referenced by:  nfexd2  2031  dvelimf  2033  nfeud2  2275  nfeud2OLD  2287  nfrald  2879  nfiotad  5485  nfixp  7385
  Copyright terms: Public domain W3C validator