Theorem exsb 1981

Description: An equivalent expression for existence. (Contributed by NM, 2-Feb-2005.)

Assertion

Ref	Expression
exsb	|- ( E. x ph <-> E. y A. x ( x = y -> ph ) )

Distinct variable groups:   x, y   ph, y

Allowed substitution hint:   ph( x)

Proof of Theorem exsb

Step	Hyp	Ref	Expression
1		ax-17 1506	. . 3 |- ( ph -> A. y ph )
2	1	sb8eh 1827	. 2 |- ( E. x ph <-> E. y [ y / x ] ph )
3		sb6 1858	. . 3 |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )
4	3	exbii 1584	. 2 |- ( E. y [ y / x ] ph <-> E. y A. x ( x = y -> ph ) )
5	2, 4	bitri 183	1 |- ( E. x ph <-> E. y A. x ( x = y -> ph ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1329   E.wex 1468   [wsb 1735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-sb 1736

This theorem is referenced by:  2exsb  1982