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Theorem rexsb 38734
Description: An equivalent expression for restricted existence, analogous to exsb 2317. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
Assertion
Ref Expression
rexsb  |-  ( E. x  e.  A  ph  <->  E. y  e.  A  A. x ( x  =  y  ->  ph ) )
Distinct variable groups:    x, y, A    ph, y
Allowed substitution hint:    ph( x)

Proof of Theorem rexsb
StepHypRef Expression
1 nfv 1769 . 2  |-  F/ y
ph
2 nfa1 1999 . 2  |-  F/ x A. x ( x  =  y  ->  ph )
3 ax12v 1951 . . 3  |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) )
4 sp 1957 . . . 4  |-  ( A. x ( x  =  y  ->  ph )  -> 
( x  =  y  ->  ph ) )
54com12 31 . . 3  |-  ( x  =  y  ->  ( A. x ( x  =  y  ->  ph )  ->  ph ) )
63, 5impbid 195 . 2  |-  ( x  =  y  ->  ( ph 
<-> 
A. x ( x  =  y  ->  ph )
) )
71, 2, 6cbvrex 3002 1  |-  ( E. x  e.  A  ph  <->  E. y  e.  A  A. x ( x  =  y  ->  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   A.wal 1450   E.wrex 2757
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-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-ex 1672  df-nf 1676  df-sb 1806  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762
This theorem is referenced by:  rexrsb  38735  2rexsb  38736
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