Theorem mo3 1440

Description: Alternate definition of "at most one." Definition of [BellMachover] p. 460, except that definition has the side condition that y not occur in ph in place of our hypothesis.

Hypothesis

Ref	Expression
mo3.1	|- (ph -> A.yph)

Assertion

Ref	Expression
mo3	|- (E*xph <-> A.xA.y((ph /\ [y / x]ph) -> x = y))

Distinct variable group:   x,y

Proof of Theorem mo3

Step	Hyp	Ref	Expression
1		mo3.1	. . 3 |- (ph -> A.yph)
2	1	mo2 1439	. 2 |- (E*xph <-> E.yA.x(ph -> x = y))
3	1	mo 1432	. 2 |- (E.yA.x(ph -> x = y) <-> A.xA.y((ph /\ [y / x]ph) -> x = y))
4	2, 3	bitri 171	1 |- (E*xph <-> A.xA.y((ph /\ [y / x]ph) -> x = y))

Syntax hints:   -> wi 3   <-> wb 144   /\ wa 221  A.wal 990   = wceq 992  E.wex 1016  [wsbc 1207  E*wmo 1420

This theorem is referenced by:  mo4f 1441  mopick 1472  isarep2 3684  gapm 11784  morex 11804

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-11 1003  ax-12 1004  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422

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