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Theorem equs45f 2088
Description: Two ways of expressing substitution when  y is not free in  ph. (Contributed by NM, 25-Apr-2008.) (Revised by Mario Carneiro, 4-Oct-2016.)
Hypothesis
Ref Expression
equs45f.1  |-  F/ y
ph
Assertion
Ref Expression
equs45f  |-  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph ) )

Proof of Theorem equs45f
StepHypRef Expression
1 equs45f.1 . . . . . 6  |-  F/ y
ph
21nfri 1778 . . . . 5  |-  ( ph  ->  A. y ph )
32anim2i 553 . . . 4  |-  ( ( x  =  y  /\  ph )  ->  ( x  =  y  /\  A. y ph ) )
43eximi 1585 . . 3  |-  ( E. x ( x  =  y  /\  ph )  ->  E. x ( x  =  y  /\  A. y ph ) )
5 equs5a 1909 . . 3  |-  ( E. x ( x  =  y  /\  A. y ph )  ->  A. x
( x  =  y  ->  ph ) )
64, 5syl 16 . 2  |-  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
)
7 equs4 1997 . 2  |-  ( A. x ( x  =  y  ->  ph )  ->  E. x ( x  =  y  /\  ph )
)
86, 7impbii 181 1  |-  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549   E.wex 1550   F/wnf 1553
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
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