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Theorem sb6f 2122
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 1778 . . . 4  |-  ( ph  ->  A. y ph )
32sbimi 1664 . . 3  |-  ( [ y  /  x ] ph  ->  [ y  /  x ] A. y ph )
4 sb4a 1948 . . 3  |-  ( [ y  /  x ] A. y ph  ->  A. x
( x  =  y  ->  ph ) )
53, 4syl 16 . 2  |-  ( [ y  /  x ] ph  ->  A. x ( x  =  y  ->  ph )
)
6 sb2 2090 . 2  |-  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph )
75, 6impbii 181 1  |-  ( [ y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549   F/wnf 1553   [wsb 1658
This theorem is referenced by:  sb5f  2123
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-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554  df-sb 1659
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