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Theorem sb6f 1979
Description: Equivalence for substitution when  y is not free in  ph. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
Hypothesis
Ref Expression
sb6f.1  |-  F/ y
ph
Assertion
Ref Expression
sb6f  |-  ( [ y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  ph )
)

Proof of Theorem sb6f
StepHypRef Expression
1 sb6f.1 . . . . 5  |-  F/ y
ph
21nfri 1742 . . . 4  |-  ( ph  ->  A. y ph )
32sbimi 1633 . . 3  |-  ( [ y  /  x ] ph  ->  [ y  /  x ] A. y ph )
4 sb4a 1864 . . 3  |-  ( [ y  /  x ] A. y ph  ->  A. x
( x  =  y  ->  ph ) )
53, 4syl 15 . 2  |-  ( [ y  /  x ] ph  ->  A. x ( x  =  y  ->  ph )
)
6 sb2 1963 . 2  |-  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph )
75, 6impbii 180 1  |-  ( [ y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527   F/wnf 1531   [wsb 1629
This theorem is referenced by:  sb5f  1980
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-ex 1529  df-nf 1532  df-sb 1630
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