Theorem exsb 2001

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 1519	. . 3 |- ( ph -> A. y ph )
2	1	sb8eh 1848	. 2 |- ( E. x ph <-> E. y [ y / x ] ph )
3		sb6 1879	. . 3 |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )
4	3	exbii 1598	. 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 1346   E.wex 1485   [wsb 1755

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-sb 1756

This theorem is referenced by:  2exsb  2002