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Theorem equsex 2143
Description: An equivalence related to implicit substitution. See equsexv 2085 for a version with a dv condition which does not require ax-13 2104. See equsexALT 2144 for an alternate proof. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Feb-2018.)
Hypotheses
Ref Expression
equsex.nf  |-  F/ x ps
equsex.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
equsex  |-  ( E. x ( x  =  y  /\  ph )  <->  ps )

Proof of Theorem equsex
StepHypRef Expression
1 equsex.nf . . 3  |-  F/ x ps
2 equsex.1 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
32biimpa 492 . . 3  |-  ( ( x  =  y  /\  ph )  ->  ps )
41, 3exlimi 2015 . 2  |-  ( E. x ( x  =  y  /\  ph )  ->  ps )
51, 2equsal 2141 . . 3  |-  ( A. x ( x  =  y  ->  ph )  <->  ps )
6 equs4 2140 . . 3  |-  ( A. x ( x  =  y  ->  ph )  ->  E. x ( x  =  y  /\  ph )
)
75, 6sylbir 218 . 2  |-  ( ps 
->  E. x ( x  =  y  /\  ph ) )
84, 7impbii 192 1  |-  ( E. x ( x  =  y  /\  ph )  <->  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376   A.wal 1450   E.wex 1671   F/wnf 1675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-12 1950  ax-13 2104
This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-nf 1676
This theorem is referenced by:  equsexh  2145  sb5rf  2271
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