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Theorem equs45f 2144
Description: Two ways of expressing substitution when  y is not free in  ph. The implication "to the left" is equs4 2088 and does not require the non-freeness hypothesis. Theorem sb56 2223 replaces the non-freeness hypothesis with a dv condition. (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 1925 . . . . 5  |-  ( ph  ->  A. y ph )
32anim2i 571 . . . 4  |-  ( ( x  =  y  /\  ph )  ->  ( x  =  y  /\  A. y ph ) )
43eximi 1702 . . 3  |-  ( E. x ( x  =  y  /\  ph )  ->  E. x ( x  =  y  /\  A. y ph ) )
5 equs5a 2033 . . 3  |-  ( E. x ( x  =  y  /\  A. y ph )  ->  A. x
( x  =  y  ->  ph ) )
64, 5syl 17 . 2  |-  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
)
7 equs4 2088 . 2  |-  ( A. x ( x  =  y  ->  ph )  ->  E. x ( x  =  y  /\  ph )
)
86, 7impbii 190 1  |-  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435   E.wex 1659   F/wnf 1663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-12 1905  ax-13 2053
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-nf 1664
This theorem is referenced by:  sb5f  2180
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