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Theorem rexsb 27269
Description: An equivalent expression for restricted existence, analogous to exsb 2135. (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 1619 . 2  |-  F/ y
ph
2 nfa1 1789 . 2  |-  F/ x A. x ( x  =  y  ->  ph )
3 ax11v 2101 . . 3  |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) )
4 sp 1748 . . . 4  |-  ( A. x ( x  =  y  ->  ph )  -> 
( x  =  y  ->  ph ) )
54com12 27 . . 3  |-  ( x  =  y  ->  ( A. x ( x  =  y  ->  ph )  ->  ph ) )
63, 5impbid 183 . 2  |-  ( x  =  y  ->  ( ph 
<-> 
A. x ( x  =  y  ->  ph )
) )
71, 2, 6cbvrex 2837 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 176   A.wal 1540   E.wrex 2620
This theorem is referenced by:  rexrsb  27270  2rexsb  27271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rex 2625
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