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Theorem equs45f 1929
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 1742 . . . . 5  |-  ( ph  ->  A. y ph )
32anim2i 552 . . . 4  |-  ( ( x  =  y  /\  ph )  ->  ( x  =  y  /\  A. y ph ) )
43eximi 1563 . . 3  |-  ( E. x ( x  =  y  /\  ph )  ->  E. x ( x  =  y  /\  A. y ph ) )
5 equs5a 1828 . . 3  |-  ( E. x ( x  =  y  /\  A. y ph )  ->  A. x
( x  =  y  ->  ph ) )
64, 5syl 15 . 2  |-  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
)
7 equs4 1899 . 2  |-  ( A. x ( x  =  y  ->  ph )  ->  E. x ( x  =  y  /\  ph )
)
86, 7impbii 180 1  |-  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527   E.wex 1528   F/wnf 1531
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
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