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Theorem nfsbd 2138
Description: Deduction version of nfsb 2136. (Contributed by NM, 15-Feb-2013.)
Hypotheses
Ref Expression
nfsbd.1  |-  F/ x ph
nfsbd.2  |-  ( ph  ->  F/ z ps )
Assertion
Ref Expression
nfsbd  |-  ( ph  ->  F/ z [ y  /  x ] ps )
Distinct variable group:    y, z
Allowed substitution hints:    ph( x, y, z)    ps( x, y, z)

Proof of Theorem nfsbd
StepHypRef Expression
1 nfsbd.1 . . . 4  |-  F/ x ph
2 nfsbd.2 . . . 4  |-  ( ph  ->  F/ z ps )
31, 2alrimi 1773 . . 3  |-  ( ph  ->  A. x F/ z ps )
4 nfsb4t 2107 . . 3  |-  ( A. x F/ z ps  ->  ( -.  A. z  z  =  y  ->  F/ z [ y  /  x ] ps ) )
53, 4syl 16 . 2  |-  ( ph  ->  ( -.  A. z 
z  =  y  ->  F/ z [ y  /  x ] ps ) )
6 a16nf 2078 . 2  |-  ( A. z  z  =  y  ->  F/ z [ y  /  x ] ps )
75, 6pm2.61d2 154 1  |-  ( ph  ->  F/ z [ y  /  x ] ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1546   F/wnf 1550   [wsb 1655
This theorem is referenced by:  nfabd2  2535
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  df-sb 1656
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