Theorem exsb 2024

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 1537	. . 3 |- ( ph -> A. y ph )
2	1	sb8eh 1866	. 2 |- ( E. x ph <-> E. y [ y / x ] ph )
3		sb6 1898	. . 3 |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )
4	3	exbii 1616	. 2 |- ( E. y [ y / x ] ph <-> E. y A. x ( x = y -> ph ) )
5	2, 4	bitri 184	1 |- ( E. x ph <-> E. y A. x ( x = y -> ph ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1362   E.wex 1503   [wsb 1773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-sb 1774

This theorem is referenced by:  2exsb  2025