Theorem exsb 2130

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

Assertion

Ref	Expression
exsb	|- (E.xph <-> E.yA.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		nfv 1619	. 2 |- F/yph
2		nfa1 1788	. 2 |- F/xA.x(x = y -> ph)
3		ax11v 2096	. . 3 |- (x = y -> (ph -> A.x(x = y -> ph)))
4		sp 1747	. . . 4 |- (A.x(x = y -> ph) -> (x = y -> ph))
5	4	com12 27	. . 3 |- (x = y -> (A.x(x = y -> ph) -> ph))
6	3, 5	impbid 183	. 2 |- (x = y -> (ph <-> A.x(x = y -> ph)))
7	1, 2, 6	cbvex 1985	1 |- (E.xph <-> E.yA.x(x = y -> ph))

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540  E.wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  2exsb  2132