Theorem exsb 1349

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-17 970	. . 3 |- (ph -> A.yph)
2	1	sb8e 1261	. 2 |- (E.xph <-> E.y[y / x]ph)
3		sb6 1266	. . 3 |- ([y / x]ph <-> A.x(x = y -> ph))
4	3	exbii 1050	. 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 953   = wceq 955  E.wex 979

This theorem is referenced by:  2exsb 1350

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171

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