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

Theorem nfald2 1912
Description: Variation on nfald 1775 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 1896 . . . . 5  |-  F/ y  -.  A. x  x  =  y
31, 2nfan 1771 . . . 4  |-  F/ y ( ph  /\  -.  A. x  x  =  y )
4 nfald2.2 . . . 4  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ps )
53, 4nfald 1775 . . 3  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x A. y ps )
65ex 423 . 2  |-  ( ph  ->  ( -.  A. x  x  =  y  ->  F/ x A. y ps ) )
7 nfa1 1756 . . 3  |-  F/ y A. y ps
8 biidd 228 . . . 4  |-  ( A. x  x  =  y  ->  ( A. y ps  <->  A. y ps ) )
98drnf1 1909 . . 3  |-  ( A. x  x  =  y  ->  ( F/ x A. y ps  <->  F/ y A. y ps ) )
107, 9mpbiri 224 . 2  |-  ( A. x  x  =  y  ->  F/ x A. y ps )
116, 10pm2.61d2 152 1  |-  ( ph  ->  F/ x A. y ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   A.wal 1527   F/wnf 1531
This theorem is referenced by:  nfexd2  1913  dvelimf  1937  nfeud2  2155  nfrald  2594  nfiotad  5222  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
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532
  Copyright terms: Public domain W3C validator