Theorem exsb 1345

Description: An equivalent expression for existence.

Assertion

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

Distinct variable groups:   x,y   ph,y

Proof of Theorem exsb

Step	Hyp	Ref	Expression
1		ax-17 968	. . 3 |- (ph -> A.yph)
2	1	sb8e 1257	. 2 |- (E.xph <-> E.y[y / x]ph)
3		sb6 1262	. . 3 |- ([y / x]ph <-> A.x(x = y -> ph))
4	3	exbii 1047	. 2 |- (E.y[y / x]ph <-> E.yA.x(x = y -> ph))
5	2, 4	bitr 173	1 |- (E.xph <-> E.yA.x(x = y -> ph))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 951   = wceq 953  E.wex 977

This theorem is referenced by:  2exsb 1346

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168

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