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Theorem nfald2 2060
Description: Variation on nfald 1871 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 2044 . . . . 5  |-  F/ y  -.  A. x  x  =  y
31, 2nfan 1846 . . . 4  |-  F/ y ( ph  /\  -.  A. x  x  =  y )
4 nfald2.2 . . . 4  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ps )
53, 4nfald 1871 . . 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 1806 . . 3  |-  F/ y A. y ps
8 biidd 229 . . . 4  |-  ( A. x  x  =  y  ->  ( A. y ps  <->  A. y ps ) )
98drnf1 2057 . . 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 1549   F/wnf 1553
This theorem is referenced by:  nfexd2  2061  dvelimf  2064  dvelimfOLD  2065  nfeud2  2292  nfrald  2749  nfiotad  5413  nfixp  7073
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554
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