Theorem exsb 1884

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 1419	. . 3 |- ( ph -> A. y ph )
2	1	sb8eh 1735	. 2 |- ( E. x ph <-> E. y [ y / x ] ph )
3		sb6 1766	. . 3 |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )
4	3	exbii 1496	. 2 |- ( E. y [ y / x ] ph <-> E. y A. x ( x = y -> ph ) )
5	2, 4	bitri 173	1 |- ( E. x ph <-> E. y A. x ( x = y -> ph ) )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   E.wex 1381   [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-sb 1646

This theorem is referenced by:  2exsb  1885