Theorem mo3h 1950

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. (Contributed by NM, 8-Mar-1995.) (New usage is discouraged.)

Hypothesis

Ref	Expression
mo3h.1	|- (ph -> A.yph)

Assertion

Ref	Expression
mo3h	|- (E*xph <-> A.xA.y((ph /\ [y / x]ph) -> x = y))

Distinct variable group:   x,y

Allowed substitution hints:   ph(x,y)

Proof of Theorem mo3h

Step	Hyp	Ref	Expression
1		mo3h.1	. . . . . . 7 |- (ph -> A.yph)
2	1	nfi 1348	. . . . . 6 |- F/yph
3	2	eu2 1941	. . . . 5 |- (E!xph <-> (E.xph /\ A.xA.y((ph /\ [y / x]ph) -> x = y)))
4	3	imbi2i 215	. . . 4 |- ((E.xph -> E!xph) <-> (E.xph -> (E.xph /\ A.xA.y((ph /\ [y / x]ph) -> x = y))))
5		df-mo 1901	. . . 4 |- (E*xph <-> (E.xph -> E!xph))
6		anclb 302	. . . 4 |- ((E.xph -> A.xA.y((ph /\ [y / x]ph) -> x = y)) <-> (E.xph -> (E.xph /\ A.xA.y((ph /\ [y / x]ph) -> x = y))))
7	4, 5, 6	3bitr4i 201	. . 3 |- (E*xph <-> (E.xph -> A.xA.y((ph /\ [y / x]ph) -> x = y)))
8		19.38 1563	. . . . 5 |- ((E.xph -> A.xA.y((ph /\ [y / x]ph) -> x = y)) -> A.x(ph -> A.y((ph /\ [y / x]ph) -> x = y)))
9	2	19.21 1472	. . . . . 6 |- (A.y(ph -> ((ph /\ [y / x]ph) -> x = y)) <-> (ph -> A.y((ph /\ [y / x]ph) -> x = y)))
10	9	albii 1356	. . . . 5 |- (A.xA.y(ph -> ((ph /\ [y / x]ph) -> x = y)) <-> A.x(ph -> A.y((ph /\ [y / x]ph) -> x = y)))
11	8, 10	sylibr 137	. . . 4 |- ((E.xph -> A.xA.y((ph /\ [y / x]ph) -> x = y)) -> A.xA.y(ph -> ((ph /\ [y / x]ph) -> x = y)))
12		anabs5 507	. . . . . 6 |- ((ph /\ (ph /\ [y / x]ph)) <-> (ph /\ [y / x]ph))
13		pm3.31 249	. . . . . 6 |- ((ph -> ((ph /\ [y / x]ph) -> x = y)) -> ((ph /\ (ph /\ [y / x]ph)) -> x = y))
14	12, 13	syl5bir 142	. . . . 5 |- ((ph -> ((ph /\ [y / x]ph) -> x = y)) -> ((ph /\ [y / x]ph) -> x = y))
15	14	2alimi 1342	. . . 4 |- (A.xA.y(ph -> ((ph /\ [y / x]ph) -> x = y)) -> A.xA.y((ph /\ [y / x]ph) -> x = y))
16	11, 15	syl 14	. . 3 |- ((E.xph -> A.xA.y((ph /\ [y / x]ph) -> x = y)) -> A.xA.y((ph /\ [y / x]ph) -> x = y))
17	7, 16	sylbi 114	. 2 |- (E*xph -> A.xA.y((ph /\ [y / x]ph) -> x = y))
18	3	simplbi2com 1330	. . 3 |- (A.xA.y((ph /\ [y / x]ph) -> x = y) -> (E.xph -> E!xph))
19	18, 5	sylibr 137	. 2 |- (A.xA.y((ph /\ [y / x]ph) -> x = y) -> E*xph)
20	17, 19	impbii 117	1 |- (E*xph <-> A.xA.y((ph /\ [y / x]ph) -> x = y))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98  A.wal 1240  E.wex 1378  [wsb 1642  E!weu 1897  E*wmo 1898

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901

This theorem is referenced by:  mo3  1951  mo2dc  1952  mo4f  1957  moim  1961  moimv  1963  moanim  1971  mopick  1975