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Theorem rexsb 27922
Description: An equivalent expression for restricted existence, analogous to exsb 2207. (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 1629 . 2  |-  F/ y
ph
2 nfa1 1806 . 2  |-  F/ x A. x ( x  =  y  ->  ph )
3 ax11v 2172 . . 3  |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) )
4 sp 1763 . . . 4  |-  ( A. x ( x  =  y  ->  ph )  -> 
( x  =  y  ->  ph ) )
54com12 29 . . 3  |-  ( x  =  y  ->  ( A. x ( x  =  y  ->  ph )  ->  ph ) )
63, 5impbid 184 . 2  |-  ( x  =  y  ->  ( ph 
<-> 
A. x ( x  =  y  ->  ph )
) )
71, 2, 6cbvrex 2929 1  |-  ( E. x  e.  A  ph  <->  E. y  e.  A  A. x ( x  =  y  ->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549   E.wrex 2706
This theorem is referenced by:  rexrsb  27923  2rexsb  27924
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-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711
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