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Theorem nfald2 2176
Description: Variation on nfald 2045 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 2163 . . . . 5  |-  F/ y  -.  A. x  x  =  y
31, 2nfan 2022 . . . 4  |-  F/ y ( ph  /\  -.  A. x  x  =  y )
4 nfald2.2 . . . 4  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ps )
53, 4nfald 2045 . . 3  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x A. y ps )
65ex 440 . 2  |-  ( ph  ->  ( -.  A. x  x  =  y  ->  F/ x A. y ps ) )
7 nfa1 1990 . . 3  |-  F/ y A. y ps
8 biidd 245 . . . 4  |-  ( A. x  x  =  y  ->  ( A. y ps  <->  A. y ps ) )
98drnf1 2174 . . 3  |-  ( A. x  x  =  y  ->  ( F/ x A. y ps  <->  F/ y A. y ps ) )
107, 9mpbiri 241 . 2  |-  ( A. x  x  =  y  ->  F/ x A. y ps )
116, 10pm2.61d2 165 1  |-  ( ph  ->  F/ x A. y ps )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 375   A.wal 1453   F/wnf 1678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1675  df-nf 1679
This theorem is referenced by:  nfexd2  2177  dvelimf  2179  nfeud2  2322  nfrald  2785  nfiotad  5568  nfixp  7567
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